Hyperbolic Geometry

on Noncommutative Balls

Gelu Popescu^{1}

Received: March 6, 2009 Revised: October 31, 2009

Communicated by Joachim Cuntz

Abstract. In this paper, we study the noncommutative balls

C_{ρ} := {(X_{1}, …, X_{n}) ∈ B(H)^{n} : ω_{ρ}(X_{1}, …, X_{n}) ≤ 1}, ρ ∈ (0, ∞],

where ω_{ρ} is the joint operator radius for n-tuples of bounded linear operators on a Hilbert space. In particular, ω_{1} is the operator norm, ω_{2} is the joint numerical radius, and ω_{∞} is the joint spectral radius.

We introduce a Harnack type equivalence relation on C_{ρ}, ρ > 0, and use it to define a hyperbolic distance δ_{ρ} on the Harnack parts (equivalence classes) of C_{ρ}. We prove that the open ball

[C_{ρ}]_{<1} := {(X_{1}, …, X_{n}) ∈ B(H)^{n} : ω_{ρ}(X_{1}, …, X_{n}) < 1}, ρ > 0,

is the Harnack part containing 0 and obtain a concrete formula for the hyperbolic distance, in terms of the reconstruction operator associated with the right creation operators on the full Fock space with n generators. Moreover, we show that the δ_{ρ}-topology and the usual operator norm topology coincide on [C_{ρ}]_{<1}. While the open ball [C_{ρ}]_{<1} is not a complete metric space with respect to the operator norm topology, we prove that it is a complete metric space with respect to the hyperbolic metric δ_{ρ}. In the particular case when ρ = 1 and H = ℂ, the hyperbolic metric δ_{ρ} coincides with the Poincaré-Bergman distance on the open unit ball of ℂ^{n}.

We introduce a Carathéodory type metric on [C_{∞}]_{<1}, the set of all n-tuples of operators with joint spectral radius strictly less then 1, by setting

d_{K}(A, B) = sup_{p} ∥ℜp(A) − ℜp(B)∥, A, B ∈ [C_{∞}]_{<1},

where the supremum is taken over all noncommutative polynomials with matrix-valued coefficients p ∈ ℂ[X_{1}, …, X_{n}]⊗M_{m}, m ∈ ℕ, with ℜp(0) = I and ℜp(X) ≥ 0 for all X ∈ C_{1}. We obtain a concrete formula for d_{K} in terms of the free pluriharmonic kernel on the noncommutative ball [C_{∞}]_{<1}. We also prove that the metric d_{K} is complete on [C_{∞}]_{<1} and its topology coincides with the operator norm topology.

1Research supported in part by an NSF grant

We provide mapping theorems, von Neumann inequalities, and Schwarz type lemmas for free holomorphic functions on noncommutative balls, with respect to the hyperbolic metric δ_{ρ}, the Carathéodory metric d_{K}, and the joint operator radius ω_{ρ}, ρ ∈ (0, ∞].

2000 Mathematics Subject Classification: Primary: 46L52; 46T25; 47A20; Secondary: 32Q45; 47A56 Keywords and Phrases: Noncommutative hyperbolic geometry; Noncommutative function theory; Harnack part; Hyperbolic distance; Carathéodory distance; Free holomorphic function; Free pluriharmonic function; Fock space; Creation operator; Joint operator radius; Joint numerical radius; Joint spectral radius; von Neumann inequality; Schwarz lemma.

Contents

Introduction 596 1. The noncommutative ball C_{ρ} and a free pluriharmonic functional

calculus 601 2. Harnack domination on noncommutative balls 608 3. Hyperbolic metric on Harnack parts of the noncommutative ball C_{ρ} 617 4. Mapping theorems for free holomorphic functions on

noncommutative balls 625 5. Carathéodory metric on the open noncommutative ball [C_{∞}]_{<1} and

Lipschitz mappings 632 6. Three metric topologies on Harnack parts of C_{ρ} 639 7. Harnack domination and hyperbolic metric for ρ-contractions (case

n = 1) 645 References 648

Introduction

In [48], we provided a generalization of the Sz.-Nagy–Foiaş theory of ρcontractions (see [54], [55], [56]), to the multivariable setting. An n-tuple (T_{1}, …, T_{n}) ∈ B(H)^{n} of bounded linear operators acting on a Hilbert space H belongs to the class C_{ρ}, ρ > 0, if there is a Hilbert space K ⊇ H and some isometries V_{i} ∈ B(K), i = 1, …, n, with orthogonal ranges such that

T_{α} = ρP_{H}V_{α}|_{H} for any α ∈ 𝔽_{n}^{+}\{g_{0}},

where P_{H} is the orthogonal projection of K onto H. Here, 𝔽_{n}^{+} stands for the unital free semigroup on n generators g_{1}, …, g_{n}, and the identity g_{0}, while T_{α} := T_{i_{1}}T_{i_{2}} ⋯ T_{i_{k}} if α = g_{i_{1}}g_{i_{2}} ⋯ g_{i_{k}} ∈ 𝔽_{n}^{+} and T_{g_{0}} := I_{H}, the identity on H. According to the theory of row contractions (see [56] for the case n = 1, and [16], [7], [32], [33], [34], for n ≥ 2) we have

C_{1} = [B(H)^{n}]_{1}^{−} := {(X_{1}, …, X_{n}) ∈ B(H)^{n} : ∥X_{1}X_{1}^{∗} + ⋯ + X_{n}X_{n}^{∗}∥^{1/2} ≤ 1} .

The results in [48] (see Section 4) can be seen as the unification of the theory of isometric dilations for row contractions [54], [56], [16], [7], [32], [33],

[34] (which corresponds to the case ρ = 1) and Berger type dilations for ntuples (T_{1}, …, T_{n}) with the joint numerical radius w(T_{1}, …, T_{n}) ≤ 1 (which corresponds to the case ρ = 2). Following the classical case ([19], [59]), we defined the joint operator radius ω_{ρ} : B(H)^{n} → [0, ∞), ρ > 0, by setting

ω_{ρ}(T_{1}, …, T_{n}) := inf \{t > 0 : (\frac{1}{t}T_{1}, …, \frac{1}{t}T_{n}) ∈ C_{ρ}\}

and ω_{∞}(T_{1}, …, T_{n}) := lim_{ρ→∞} ω_{ρ}(T_{1}, …, T_{n}). In particular, ω_{1}(T_{1}, …, T_{n}) coin-

cides with the norm of the row operator [T_{1} ⋯ T_{n}], ω_{2}(T_{1}, …, T_{n}) coincides with the joint numerical radius w(T_{1}, …, T_{n}), and ω_{∞}(T_{1}, …, T_{n}) is equal to the (algebraic) joint spectral radius (see [7], [25])

r(T_{1}, …, T_{n}) := lim_{k→∞} ∥ ∑_{|α|=k} T_{α}T_{α}^{∗}∥^{1/2k} ,

where the length of α ∈ 𝔽_{n}^{+} is defined by |α| := 0 if α = g_{0} and by |α| := k if α = g_{i_{1}} ⋯ g_{i_{k}} and i_{1}, …, i_{k} ∈ {1, …, n}. In [48], we considered basic properties of the joint operator radius ω_{ρ} and we extended to the (noncommutative and commutative) multivariable setting several classical results obtained by Sz.- Nagy and Foiaş, Halmos, Berger and Stampfli, Holbrook, Paulsen, Badea and Cassier, and others (see [2], [3], [4], [5], [17], [18], [19], [20], [21], [29], [30], [55], and [59]). In [49], we introduced a hyperbolic metric δ on the open noncommutative ball [B(H)^{n}]_{1}, which turned out to be a noncommutative extension of the Poincaré-Bergman ([6]) metric on the open unit ball 𝔹_{n} := {z ∈ ℂ^{n} : ∥z∥_{2} < 1}. We proved that δ is invariant under the action of the group Aut([B(H)^{n}]_{1}) of all free holomorphic automorphisms of [B(H)^{n}]_{1}, and showed that the δ-topology and the usual operator norm topology coincide on [B(H)^{n}]_{1}. Moreover, we proved that [B(H)^{n}]_{1} is a complete metric space with respect to the hyperbolic metric and obtained an explicit formula for δ in terms of the reconstruction operator. A Schwarz-Pick lemma for bounded free holomorphic functions on [B(H)^{n}]_{1}, with respect to the hyperbolic metric, was also obtained. In [46], we continued to study the noncommutative hyperbolic geometry on the unit ball of B(H)^{n}, its connections with multivariable dilation theory, and its implications to noncommutative function theory. The results from [49] and [46] make connections between noncommutative function theory (see [41], [44], [50], [47]) and classical results in hyperbolic complex analysis (see [22], [23], [24], [52], [58]).

The present paper is an attempt to extend the results [49] concerning the noncommutative hyperbolic geometry of the unit ball [B(H)^{n}]_{1} to the more general setting of [48]. We study the noncommutative balls

[C_{ρ}]_{<1} = {(X_{1}, …, X_{n}) ∈ B(H)^{n} : ω_{ρ}(X_{1}, …, X_{n}) < 1} , ρ ∈ (0, ∞],

and the Harnach parts of C_{ρ}, ρ > 0, as metric spaces with respect to a hyperbolic (resp. Carathéodory) type metric that will be introduced. We provide mapping theorems for free holomorphic functions on these noncommutative balls, extending classical results from complex analysis and hyperbolic geometry.

In Section 1, we consider some preliminaries on free holomorphic (resp. pluriharmonic) functions on the open unit ball [B(H)^{n}]_{1}, and present several characterizations for the n-tuples of operators of class C_{ρ}, ρ ∈ (0, ∞). We introduce a free pluriharmonic functional calculus for the class C_{ρ} and show that a von Neumann type inequality characterizes this class. In particular, we prove that an n-tuple of operators (T_{1}, …, T_{n}) ∈ B(H)^{n} is of class C_{ρ} if and only if

∥p(T_{1}, …, T_{n})∥ ≤ ∥ρp(S_{1}, …, S_{n}) + (1 − ρ)p(0)∥

for any noncommutative polynomial with matrix-valued coefficients p ∈ ℂ[Z_{1}, …, Z_{n}] ⊗ M_{m}, m ∈ ℕ, where S_{1}, …, S_{n} are the left creation operators on the full Fock space with n generators.

In Section 2, we introduce a preorder relation ≺^{H} on the class C_{ρ}. If A := (A_{1}, …, A_{n}) and B := (B_{1}, …, B_{n}) are in the class C_{ρ} ⊂ B(H)^{n}, we say that

A is Harnack dominated by B (denote A ≺^{H} B) if there exists c > 0 such that

ℜp(A_{1}, …, A_{n}) + (ρ − 1)ℜp(0) ≤ c^{2} [ℜp(B_{1}, …, B_{n}) + (ρ − 1)ℜp(0)]

for any noncommutative polynomial with matrix-valued coefficients p ∈ ℂ[X_{1}, …, X_{n}] ⊗ M_{m}, m ∈ ℕ, such that ℜp(X) := \frac{1}{2}[p(X)^{∗} + p(X)] ≥ 0 for any X ∈ [B(K)^{n}]_{1}, where K is an infinite dimensional Hilbert space. When we

want to emphasize the constant c, we write A≺_{c}^{H} B. We provide several char-

acterizations for the Harnack domination on the noncommutative ball C_{ρ} (see Theorem 2.2), and determine the set of all elements in C_{ρ} which are Harnack dominated by 0. The results of this section will play a major role in the next sections.

The relation ≺^{H} induces an equivalence relation ∼^{H} on the class C_{ρ}. More precisely, two n-tuples A and B are Harnack equivalent (and denote A ∼^{H} B)

if and only if there exists c > 1 such that A≺_{c}^{H} B and B≺_{c}^{H} A (in this case we

denote A∼_{c}^{H} B). The equivalence classes with respect to ∼^{H} are called Harnack parts of C_{ρ}. In Section 3, we provide a Harnack type double inequality for positive free pluriharmonic functions on the noncommutative ball C_{ρ} and use it to prove that the Harnack part of C_{ρ} which contains 0 coincides with the open noncommutative ball

[C_{ρ}]_{<1} := {(X_{1}, …, X_{n}) ∈ B(H)^{n} : ω_{ρ}(X_{1}, …, X_{n}) < 1}.

We introduce a hyperbolic metric δ_{ρ} : Δ × Δ → ℝ^{+} on any Harnack part Δ of C_{ρ}, by setting

δ_{ρ}(A, B) := ln inf {c > 1 : A ∼_{c}^{H} B} , A, B ∈ Δ.

A concrete formula for the hyperbolic distance on any Harnack part of C_{ρ} is obtained. When Δ = [C_{ρ}]_{<1}, we prove that

δ_{ρ}(A, B) = ln max {∥C_{ρ,A}C_{ρ,B}^{−1}∥ , ∥C_{ρ,B}C_{ρ,A}^{−1}∥} , A, B ∈ [C_{ρ}]_{<1},

where

\begin{aligned}C_{ρ,X} & := Δ_{ρ,X}(I − R_{X})^{−1}, \\ Δ_{ρ,X} & := [ρI + (1 − ρ)(R_{X}^{∗} + R_{X}) + (ρ − 2)R_{X}^{∗}R_{X}]^{1/2} , \end{aligned}

and R_{X} := X_{1}^{∗}⊗R_{1}+⋯+X_{n}^{∗}⊗R_{n} is the reconstruction operator associated with the right creation operators R_{1}, …, R_{n} on the full Fock space with n generators, and X := (X_{1}, …, X_{n}) ∈ [C_{ρ}]_{<1}. We recall that the reconstruction operator has played an important role in noncommutative multivariable operator theory. It appeared as a building block in the characteristic function associated to a row contraction (see [34], [45]) and also as a quantized variable (associated with the n-tuple X) in the noncommutative Cauchy, Poisson, and Berezin transform, respectively (see [41], [44], [47], [48]). In Section 4, we study the stability of the ball C_{ρ} under contractive free holomorphic functions and provide mapping theorems, von Neumann inequalities, and Schwarz type lemmas, with respect to the hyperbolic metric δ_{ρ} and the operator radius ω_{ρ}, ρ ∈ (0, ∞]. Let f := (f_{1}, …, f_{m}) be a contractive free holomorphic function with ∥f(0)∥ < 1 such that the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n} (see [36], [40]). If an n-tuple of operators (T_{1}, …, T_{n}) ∈ B(H)^{n} is of class C_{ρ}, ρ > 0, then we prove that, under the free pluriharmonic functional calculus, the m-tuple f(T_{1}, …, T_{n}) ∈ B(H)^{m} is of class C_{ρ_{f}}, where ρ_{f} > 0 is given in terms of ρ and f(0). One of the main results of this section is the following spectral von Neumann inequality for n-tuples of operators. If f := (f_{1}, …, f_{m}) satisfies the conditions above and (T_{1}, …, T_{n}) ∈ B(H)^{n} has the joint spectral radius r(T_{1}, …, T_{n}) < 1, then r(f(T_{1}, …, T_{n})) < 1. If, in addition, f(0) = 0 and δ_{ρ} : Δ × Δ → [0, ∞) is the hyperbolic metric on a Harnack part Δ of C_{ρ}, then we prove that

δ_{ρ}(f(A), f(B)) ≤ δ_{ρ}(A, B), A, B ∈ Δ.

In particular, this holds when Δ is the open ball [C_{ρ}]_{<1}. Moreover, in this setting, we show that

ω_{ρ}(f(T_{1}, …, T_{n})) < 1, (T_{1}, …, T_{n}) ∈ [C_{ρ}]_{<1},

for any ρ > 0. The general case when f(0) ≠ 0 is also discussed. In Section 5, we introduce a Carathéodory type metric on the set of all n-tuples of operators with joint spectral radius strictly less then 1, i.e.,

[C_{∞}]_{<1} := {(X_{1}, …, X_{n}) ∈ B(H)^{n} : r(X_{1}, …, X_{n}) < 1},

by setting

d_{K}(A, B) = sup_{p} ∥ℜp(A) − ℜp(B)∥,

where the supremum is taken over all noncommutative polynomials with matrix-valued coefficients p ∈ ℂ[X_{1}, …, X_{n}] ⊗ M_{m}, m ∈ ℕ, with ℜp(0) = I and ℜp(X) ≥ 0 for all X ∈ [B(K)^{n}]_{1}. We obtain a concrete formula for d_{K} in terms of the free pluriharmonic kernel on the open unit ball [C_{∞}]_{<1}. More precisely, we show that

d_{K}(A, B) = ∥P (A, R) − P (B, R)∥, A, B ∈ [C_{∞}]_{<1},

where

P (X, R) := ∑_{k=1}^{∞} ∑_{|α|=k} X_{α} ⊗ R∗ ; α̃ + ρI ⊗ I + ∑_{k=1}^{∞} ∑_{|α|=k} X_{α}^{∗} ⊗ Rα̃, X ∈ [C_{∞}]_{<1},

and α̃ is the reverse of α ∈ 𝔽_{n}^{+}. This is used to prove that the metric d_{K} is complete on [C_{∞}]_{<1} and its topology coincides with the operator norm topology. We also prove that if f := (f_{1}, …, f_{m}) is a contractive free holomorphic function with ∥f(0)∥ < 1 such that the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n}, then

d_{K}(f(A), f(B)) ≤ \frac{1 + ∥f(0)∥}{1 − ∥f(0)∥}d_{K}(A, B), A, B ∈ [C_{∞}]_{<1}.

As a consequence, we deduce that the map

[C_{∞}]_{<1} ∋ (X_{1}, …, X_{n}) ↦ f(X_{1}, …, X_{n}) ∈ [C_{∞}]_{<1}

is continuous in the operator norm topology. In Section 6, we compare the hyperbolic metric δ_{ρ} with the Carathéodory metric d_{K}, and the operator metric, respectively, on Harnack parts of the unit ball C_{ρ}, ρ > 0. In particular, we prove that the hyperbolic metric δ_{ρ} is complete on the open unit unit ball [C_{ρ}]_{<1}, while the other two metrics, mentioned above, are not complete. On the other hand, we show the δ_{ρ}-topology, the d_{K}-topology, and the operator norm topology coincide on [C_{ρ}]_{<1}. In Section 7, we consider the single variable case (n = 1) and show that our Harnack domination for ρ-contractions is equivalent to the one introduced and studied by G. Cassier and N. Suciu in [9] and [10]. Consequently, we recover some of their results and, moreover, we obtain some results which seem to be new even in the single variable case. Finally, we want to acknowledge that we were influenced in writing this paper by the work of C. Foiaş ([15]), I. Suciu ([53]), and G. Cassier and N. Suciu ([9], [10]) concerning the Harnack domination and the hyperbolic distance between two ρ-contractions. It will be interesting to see to which extent the results of this paper, concerning the hyperbolic geometry on noncommutative balls, can be extended to the Hardy algebras of Muhly and Solel (see [26], [27], [28]).

1. The noncommutative ball C_{ρ} and a free pluriharmonic functional calculus

In this section, we consider some preliminaries on free holomorphic (resp. pluriharmonic) functions on the unit ball [B(H)^{n}]_{1}, and several characterizations for the n-tuples of operators of class C_{ρ}. We introduce a free pluriharmonic functional calculus for the class C_{ρ} and show that a von Neumann type inequality characterizes the class C_{ρ}. Let H_{n} be an n-dimensional complex Hilbert space with orthonormal basis e_{1}, e_{2}, …, e_{n}, where n = 1, 2, …, or n = ∞. The full Fock space of H_{n} is defined by

F^{2}(H_{n}) := ℂ1 ⊕ ⨁_{k≥1} H_{n}^{⊗k},

where H_{n}^{⊗k} is the (Hilbert) tensor product of k copies of H_{n}. We define the left (resp. right) creation operators S_{i} (resp. R_{i}), i = 1, …, n, acting on the full Fock space F^{2}(H_{n}) by setting

S_{i}φ := e_{i} ⊗ φ, φ ∈ F^{2}(H_{n}),

(resp. R_{i}φ := φ ⊗ e_{i}, φ ∈ F^{2}(H_{n})). We recall that the noncommutative disc algebra A_{n} (resp. R_{n}) is the norm closed algebra generated by the left (resp. right) creation operators and the identity. The noncommutative analytic Toeplitz algebra F_{n}^{∞} (resp. R_{n}^{∞}) is the weakly closed version of A_{n} (resp. R_{n}). These algebras were introduced in [36] in connection with a von Neumann type inequality [57], as noncommutative analogues of the disc algebra A(𝔻) and the Hardy space H^{∞}(𝔻). For more information on theses noncommutative algebras we refer the reader to [35], [37], [38], [40], [12], and the references therein. Let H be a Hilbert space and let B(H) be the algebra of all bounded linear operators on H. We identify M_{m}(B(H)), the set of m×m matrices with entries from B(H), with B(H^{(m)}), where H^{(m)} is the direct sum of m copies of H. If X is an operator space, i.e., a closed subspace of B(H), we consider M_{m}(X) as a subspace of M_{m}(B(H)) with the induced norm. Let X , Y be operator spaces and u : X → Y be a linear map. Define the map u_{m} : M_{m}(X) → M_{m}(Y) by

u_{m}([x_{ij}]) := [u(x_{ij})].

We say that u is completely bounded if

∥u∥_{cb} := sup_{m≥1} ∥u_{m}∥ < ∞.

If ∥u∥_{cb} ≤ 1 (resp. u_{m} is an isometry for any m ≥ 1) then u is completely contractive (resp. isometric), and if u_{m} is positive for all m, then u is called completely positive. For basic results concerning completely bounded maps and operator spaces we refer to [29], [31], and [13]. A few more notations and definitions are necessary. If ω, γ ∈ 𝔽_{n}^{+}, we say that ω >_{l} γ if there is σ ∈ 𝔽_{n}^{+}\{g_{0}} such that ω = γσ and set ω\_{l}γ := σ. We denote by α̃ the reverse of α ∈ 𝔽_{n}^{+}, i.e., α̃ = g_{i_{k}} ⋯ g_{i_{1}} if α = g_{i_{1}} ⋯ g_{i_{k}} ∈ 𝔽_{n}^{+}. An operator-valued positive semidefinite kernel on the free semigroup 𝔽_{n}^{+} is a map K : 𝔽_{n}^{+} × 𝔽_{n}^{+} → B(H) with the property that for each k ∈ ℕ, for each choice of vectors h_{1}, …, h_{k} in H, and σ_{1}, …, σ_{k} in 𝔽_{n}^{+}, the inequality

∑_{i,j=1}^{k}⟨K(σ_{i}, σ_{j})h_{j}, h_{i}⟩ ≥ 0

holds. Such a kernel is called multi-Toeplitz if it has the following properties: K(α, α) = I_{H} for any α ∈ 𝔽_{n}^{+}, and

K(σ, ω) = \begin{cases}K(g_{0}, ω\_{l}σ) & if ω >_{l} σ \\ K(σ\_{l}ω, g_{0}) & if σ >_{l} ω \\ 0 & otherwise. \end{cases}

An n-tuple of operators (T_{1}, …, T_{n}), T_{i} ∈ B(H), belongs to the class C_{ρ}, ρ > 0, if there exist a Hilbert space K ⊇ H and isometries V_{i} ∈ B(K), i = 1, …, n, with orthogonal ranges, such that

T_{α} = ρP_{H}V_{α}|_{H}, α ∈ 𝔽_{n}^{+}\{g_{0}},

where P_{H} is the orthogonal projection of K onto H. If K = K_{T} := ⋁_{α∈𝔽_{n}^{+}} V_{α}H, then the n-tuple (V_{1}, …, V_{n}) is the minimal isometric dilation of (T_{1}, …, T_{n}), which is unique up to an isomorphism. Note that if (T_{1}, …, T_{n}) ∈ C_{ρ}, then the joint spectral radius r(T_{1}, …, T_{n}) ≤ 1, where

r(T_{1}, …, T_{n}) := lim_{k→∞} ∥ ∑_{|α|=k} T_{α}T_{α}^{∗}∥^{1/2k} .

We recall (see Corollary 1.36 from [48]) that ⋃_{ρ>0} C_{ρ} is dense (in the operator

norm topology) in the set of all n-tuples of operators with joint spectral radius r(T_{1}, …, T_{n}) ≤ 1. Moreover, any n-tuple of operators with r(T_{1}, …, T_{n}) < 1 is of class C_{ρ} for some ρ > 0. We should add that (see Theorem 5.9 from [43])

(T_{1}, …, T_{n}) ∈ B(H)^{n} has the joint spectral radius r(T_{1}, …, T_{n}) < 1 if and only if it is uniformly stable, i.e., ∥ ∑_{|α|=k} T_{α}T_{α}^{∗}∥ → 0, as k → ∞.

Since the joint spectral radius of n-tuples of operators plays an important role in the present paper, we recall (see [7], [25]) some of its properties. The joint right spectrum σ_{r}(T_{1}, …, T_{n}) of an n-tuple (T_{1}, …, T_{n}) of operators in B(H) is the set of all n-tuples (λ_{1}, …, λ_{n}) of complex numbers such that the right ideal of B(H) generated by the operators λ_{1}I − T_{1}, …, λ_{n}I − T_{n} does not contain the identity operator. We know that σ_{r}(T_{1}, …, T_{n}) is included in the closed ball of ℂ^{n} of radius r(T_{1}, …, T_{n}).

If we assume that T_{1}, …, T_{n} ∈ B(H) are mutually commuting operators and B is a closed subalgebra of B(H) containing T_{1}, …, T_{n}, and the identity, then the Harte spectrum σ(T_{1}, …, T_{n}) is the set of all (λ_{1}, …, λ_{n}) ∈ ℂ^{n} such that

(λ_{1}I − T_{1})X_{1} + ⋯ + (λ_{n}I − T_{n})X_{n} ≠ I

for all X_{1}, …, X_{n} ∈ B. In this case, we have

r(T_{1}, …, T_{n}) = max{∥(λ_{1}, …, λ_{n})∥_{2} : (λ_{1}, …, λ_{n}) ∈ σ(T_{1}, …, T_{n})}.

According to [25], the latter formula remains true if the Harte spectrum is replaced by the Taylor’s spectrum for commuting operators.

According to Theorem 4.1 from [39] and Theorems 1.34 and 1.39 from [48], we have the following characterizations for the n-tuples of operators of class C_{ρ}. We denote by ℂ[Z_{1}, …, Z_{n}] the set of all noncommutative polynomials in n noncommuting indeterminates.

Theorem 1.1. Let T_{1}, …, T_{n} ∈ B(H) and let S ⊂ C^{∗}(S_{1}, …, S_{n}) be the operator system defined by

S := {p(S_{1}, …, S_{n}) + q(S_{1}, …, S_{n})^{∗} : p, q ∈ ℂ[Z_{1}, …, Z_{n}]}.

Then the following statements are equivalent:

(i) (T_{1}, …, T_{n}) ∈ C_{ρ}.

(ii) The map Ψ : S → B(H) defined by

\begin{aligned}Ψ (p(S_{1}, …, S_{n}) + q(S_{1}, …, S_{n})^{∗}) := p(T_{1}, …, T_{n}) & + q(T_{1}, …, T_{n})^{∗} \\ & + (ρ − 1)(p(0) + \overline{q(0)})I \end{aligned}

is completely positive. (iii) The joint spectral radius r(T_{1} …, T_{n}) ≤ 1 and the ρ-pluriharmonic

kernel defined by

P_{ρ}(rT, R) := ∑_{k=1}^{∞} ∑_{|α|=k} r^{|α|}T_{α} ⊗ R∗ ; α̃ + ρI ⊗ I + ∑_{k=1}^{∞} ∑_{|α|=k} r^{|α|}T_{α}^{∗} ⊗ Rα̃

is positive for any 0 < r < 1, where the convergence is in the operator norm topology.

(iv) The spectral radius r(T_{1}, …, T_{n}) ≤ 1 and

ρI ⊗ I + (1 − ρ)r ∑_{i=1}^{n}(T_{i} ⊗ R_{i}^{∗} + T_{i}^{∗} ⊗ R_{i}) + (ρ − 2)r^{2} (∑_{i=1}^{n} T_{i}T_{i}^{∗} ⊗ I) ≥ 0

for any 0 < r < 1.

(v) The multi-Toeplitz kernel K_{ρ,T} : 𝔽_{n}^{+} × 𝔽_{n}^{+} → B(H) defined by

K_{ρ,T} (α, β) := \begin{cases}\frac{1}{ρ}T_{β\_{l}α} & if β >_{l} α \\ I & if α = β \\ \frac{1}{ρ}(T_{α\_{l}β})^{∗} & if α >_{l} β \\ 0 & otherwise \end{cases}

is positive semidefinite.

Consider 1 ≤ m < n and let (R′_{1}, …, R′_{m}) and (R_{1}, …, R_{n}) be the right creation operators on F^{2}(H_{m}) and F^{2}(H_{n}), respectively. According to the Wold type decomposition for isometries with orthogonal ranges [33], the mtuple (R_{1}, …, R_{m}) is unitarily equivalent to (R′_{1} ⊗I_{E}, …, R′_{m} ⊗I_{E}), where E is equal to F^{2}(H_{n}) ⊖ F^{2}(H_{m}). Consequently, using Theorem 1.1, one can easily deduce the following result.

Corollary 1.2. Let ρ > 0, 1 ≤ m < n, and consider an m-tuple (T_{1}, …, T_{m}) ∈ B(H)^{m} and its extension (T_{1}, …, T_{m}, 0, …, 0) ∈ B(H)^{n}. Then the following statements hold:

(i) (T_{1}, …, T_{m}) ∈ C_{ρ} if and only if (T_{1}, …, T_{m}, 0, …, 0) ∈ C_{ρ}; (ii) ω_{ρ}(T_{1}, …, T_{m}) = ω_{ρ}(T_{1}, …, T_{m}, 0, …, 0)); (iii) r(T_{1}, …, T_{m}) = r(T_{1}, …, T_{m}, 0, …, 0).

Throughout this paper, we assume that E is a separable Hilbert space. We recall [44] that a mapping F : [B(H)^{n}]_{1} → B(H) ¯⊗_{min}B(E) is called free holomorphic function on [B(H)^{n}]_{1} with coefficients in B(E) if there exist A_{(α)} ∈ B(E),

α ∈ 𝔽_{n}^{+}, such that lim sup_{k→∞} ∥∑_{|α|=k} A_{(α)}^{∗}A_{(α)}∥^{1/2k} ≤ 1 and

F (X_{1}, …, X_{n}) = ∑_{k=0}^{∞} ∑_{|α|=k} X_{α} ⊗ A_{(α)},

where the series converges in the operator norm topology for any (X_{1}, …, X_{n}) in the open unit ball [B(H)^{n}]_{1} := {(X_{1}, …, X_{n}) : ∥X_{1}X_{1}^{∗} +⋯+X_{n}X_{n}∥ < 1}. The set of all free holomorphic functions on [B(H)^{n}]_{1} with coefficients in B(E) is denoted by H_{ball}(B(E)). Let H_{ball}^{∞}(B(E)) denote the set of all elements F in H_{ball}(B(E)) such that

∥F ∥_{∞} := sup ∥F (X_{1}, …, X_{n})∥ < ∞,

where the supremum is taken over all n-tuples of operators (X_{1}, …, X_{n}) ∈ [B(H)^{n}]_{1} and any Hilbert space H. According to [44] and [47], H_{ball}^{∞}(B(E)) can be identified to the operator algebra F_{n}^{∞} ¯⊗B(E) (the weakly closed algebra generated by the spatial tensor product), via the noncommutative Poisson transform. Due to the fact that a free holomorphic function is uniquely determined by its representation on an infinite dimensional Hilbert space, we identify, throughout this paper, a free holomorphic function with its representation on a separable infinite dimensional Hilbert space.

We say that a map u : [B(H)^{n}]_{1} → B(H) ¯⊗_{min}B(E) is a self-adjoint free pluriharmonic function on [B(H)^{n}]_{1} if u = ℜf := \frac{1}{2}(f^{∗} + f) for some free holomorphic function f. A free pluriharmonic function on [B(H)^{n}]_{1} has the form H := H_{1} + iH_{2}, where H_{1}, H_{2} are self-adjoint free pluriharmonic functions on [B(H)^{n}]_{1}. We recall [47] that if

f(Z_{1}, …, Z_{n}) = ∑_{k=1}^{∞} ∑_{|α|=k} Z_{α}^{∗} ⊗ B_{(α)} + I ⊗ A_{(0)} + ∑_{k=1}^{∞} ∑_{|α|=k} Z_{α} ⊗ A_{(α)}

is a free pluriharmonic function on [B(H)^{n}]_{1} with coefficients in B(E) and (T_{1}, …, T_{n}) ∈ B(H)^{n} is any n-tuple of operators with joint spectral radius r(T_{1}, …, T_{n}) < 1, then f(T_{1}, …, T_{n}) is a bounded linear operator, where the corresponding series converge in norm. Moreover lim_{r→1} f(rT_{1}, …, rT_{n}) = f(T_{1}, … T_{n}) in the operator norm topology. We refer to [47] for more results on free pluriharmonic functions.

We denote by Har_{ball}^{c}(B(E)) the set of all free pluriharmonic functions on [B(H)^{n}]_{1} with operator-valued coefficients in B(E), which have continuous extensions (in the operator norm topology) to the closed ball [B(H)^{n}]_{1}^{−}. We assume that H is an infinite dimensional Hilbert space. According to Theorem 4.1 from [47], we can identify Har_{ball}^{c}(B(E)) with the operator space

\overline{A_{n}(E)^{∗} + A_{n}(E)}^{∥·∥}, where A_{n}(E) := A_{n} ¯⊗_{min}B(E) and A_{n} is the noncommutative disc algebra. More precisely, if u : [B(H)^{n}]_{1} → B(H) ¯⊗_{min}B(E), then the following statements are equivalent:

(a) u is a free pluriharmonic function on [B(H)^{n}]_{1} which has a continuous extension (in the operator norm topology) to the closed ball [B(H)^{n}]_{1}^{−};

(b) there exists f ∈ \overline{A_{n}(E)^{∗} + A_{n}(E)}^{∥·∥} such that u(X) = (P_{X} ⊗ id)(f) for X ∈ [B(H)^{n}]_{1}, where P_{X} is the noncommutative Poisson transform at X;

(c) u is a free pluriharmonic function on [B(H)^{n}]_{1} such that u(rS_{1}, …, rS_{n}) converges in the operator norm topology, as r → 1.

In this case, we have f = lim_{r→1} u(rS_{1}, …, rS_{n}), where the convergence is

in the operator norm topology. Moreover, the map Φ : Har_{ball}^{c}(B(E)) →

\overline{A_{n}(E)^{∗} + A_{n}(E)}^{∥·∥} defined by Φ(u) := f is a completely isometric isomorphism of operator spaces. We call f the model boundary function of u.

Now, we introduce a free pluriharmonic functional calculus for the class C_{ρ}.

Theorem 1.3. Let T := (T_{1}, …, T_{n}) ∈ B(H)^{n} be of class C_{ρ}, and let u ∈ Har_{ball}^{c}(B(E)) have the standard representation

u(X_{1}, …, X_{n}) = ∑_{k=1}^{∞} ∑_{|α|=k} X_{α}^{∗} ⊗ B_{(α)} + I ⊗ A_{(0)} + ∑_{k=1}^{∞} ∑_{|α|=k} X_{α} ⊗ A_{(α)}

on [B(H)^{n}]_{1}, for some A_{(α)}, B_{(α)} ∈ B(E), where the series converge in the operator norm topology. Then

u(T_{1}, …, T_{n}) := lim_{r→1} u(rT_{1}, …, rT_{1})

exists in the operator norm and

∥u(T_{1}, …, T_{n})∥ ≤ ∥ρu + (1 − ρ)u(0)∥_{∞}.

Proof. Since T := (T_{1}, …, T_{n}) ∈ B(H)^{n} is an n-tuple of class C_{ρ}, there is a minimal isometric dilation V := (V_{1}, …, V_{n}) of T on a Hilbert space K_{T} ⊇ H, satisfying the following properties: V_{i}^{∗}V_{j} = δ_{ij}I for i, j = 1, …, n, and T_{α} = ρP_{H}V_{α}|_{H} for any α ∈ 𝔽_{n}^{+}\{g_{0}}, and K_{T} = ⋁_{α∈𝔽_{n}^{+}} V_{α}H. Taking into account that u ∈ Har_{ball}^{c}(B(E)), we have

u(rV_{1}, …, rV_{n}) = ∑_{k=1}^{∞} ∑_{|α|=k} r^{|α|}V_{α}^{∗} ⊗ B_{(α)} + I ⊗ A_{(0)} + ∑_{k=1}^{∞} ∑_{|α|=k} r^{|α|}V_{α} ⊗ A_{(α)},

where the convergence is in the operator norm. Hence, and due to the fact that

∑_{|α|=k} r^{|α|}T_{α}^{∗} ⊗ B_{(α)} = ρ(P_{H} ⊗ I) ( ∑_{|α|=k} r^{|α|}V_{α}^{∗} ⊗ B_{(α)}) |_{H⊗E}, k = 1, 2, …,

we deduce that

\begin{aligned}u(rT_{1}, …, rT_{n}) & := ∑_{k=1}^{∞} ∑_{|α|=k} r^{|α|}T_{α}^{∗} ⊗ B_{(α)} + I ⊗ A_{(0)} + ∑_{k=1}^{∞} ∑_{|α|=k} r^{|α|}T_{α} ⊗ A_{(α)} \\ & = ρ(P_{H} ⊗ I)u(rV_{1}, …, rV_{n})|_{H⊗E} − (ρ − 1)u(0). \end{aligned}

exists in the operator norm topology. Now, taking into account that lim_{r→1} u(rV_{1}, …, rV_{1}) exists in the operator norm, we deduce that lim_{r→1} u(rT_{1}, …, rT_{1}) exists in the same topology. Consequently, we can define

u(T_{1}, …, T_{n}) := lim_{r→1} u(rT_{1}, …, rT_{1}).

Using the considerations above, and the noncommutative von Neumann inequality, we obtain

∥u(T_{1}, …, T_{n})∥ ≤ ∥ρu + (1 − ρ)u(0)∥_{∞} ≤ (ρ + |ρ − 1|)∥u∥_{∞}

for any (T_{1}, …, T_{n}) ∈ C_{ρ}. □

We will refer to the map

Har_{ball}^{c}(B(E)) ∋ u ↦ u(T_{1}, …, T_{n}) ∈ B(H) ¯⊗_{min}B(E)

as the free pluriharmonic functional calculus for the class C_{ρ}. Since there is

a completely isometric isomorphism of operator spaces \overline{A_{n}(E)^{∗} + A_{n}(E)}^{∥·∥} ∋ f ↦ u ∈ Har_{ball}^{c}(B(E)), given by u = (P_{X} ⊗ id)(f) for X ∈ [B(H)^{n}]_{1}, we also use the notation f(T_{1}, …, T_{n}) for u(T_{1}, …, T_{n}). Now, we show that the von Neumann type inequality of Theorem 1.3 characterizes the class C_{ρ}. Denote

P(S_{1}, …, S_{n}) := {p(S_{1}, …, S_{n}) : p ∈ ℂ[Z_{1}, …, Z_{n}]},

where S_{1}, …, S_{n} are the left creation operators on the full Fock space F^{2}(H_{n}).

Theorem 1.4. Let T := (T_{1}, …, T_{n}) ∈ B(H)^{n} be an n-tuple of operators. Then the following statements are equivalent:

(i) T is of class C_{ρ}; (ii) the von Neumann type inequality

∥p(T_{1}, …, T_{n})∥ ≤ ∥ρp(S_{1}, …, S_{n}) + (1 − ρ)p(0)∥

holds for any noncommutative polynomial p ∈ ℂ[Z_{1}, …, Z_{n}] ⊗ M_{m}, m ∈ ℕ;

(iii) the map Ψ_{T} : A_{n} → B(H) defined by

Ψ_{T} (q(S_{1}, …, S_{n})) := \frac{1}{ρ}q(T_{1}, …, T_{n}) + (1 − \frac{1}{ρ}) q(0)I

for q(S_{1}, …, S_{n}) ∈ P(S_{1}, …, S_{n}) is completely contractive.

Proof. The implication (i) ⟹ (ii) follows, in particular, from Theorem 1.3. To prove the implication (ii) ⟹ (iii), note that setting p := \frac{1}{ρ}q +

(1 − \frac{1}{ρ}) q(0)I, where q ∈ ℂ[Z_{1}, …, Z_{n}] ⊗ M_{m}, m ∈ ℕ, we have

∥Ψ_{T} (q(S_{1}, …, S_{n}))∥ = ∥p(T_{1}, …, T_{n})∥
≤ ∥ρp(S_{1}, …, S_{n}) + (1 − ρ)p(0)∥
= ∥q(S_{1}, …, S_{n})∥,

which proves that Ψ_{T} is completely contractive on the set of all polynomials P(S_{1}, …, S_{n}) and, consequently, extends uniquely to a completely contractive map on the noncommutative disc algebra A_{n}. It remains to prove that (iii) ⟹ (i). Due to Arveson’s extension theorem, item (iii) implies the existence of a unique completely positive extension Ψ̃_{T} : A_{n}^{∗} + A_{n} → B(H) of Ψ_{T}. Note that

Ψ̃_{T} (r(S_{1}, …, S_{n}) + q(S_{1}, …, S_{n})^{∗}) =

= \frac{1}{ρ}(r(T_{1}, …, T_{n}) + q(T_{1}, …, T_{n})^{∗}) + (1 − \frac{1}{ρ}) (r(0) + \overline{q(0)})I

for any polynomials r(S_{1}, …, S_{n}) and q(S_{1}, …, S_{n}) in P(S_{1}, …, S_{n}). Applying Theorem 1.1 (the equivalence (i) ↔ (ii)), we complete the proof. □

2. Harnack domination on noncommutative balls

We introduce a preorder relation ≺^{H} on the noncommutative ball C_{ρ}, ρ ∈ (0, ∞), and provide several characterizations. We determine the elements of C_{ρ} which are Harnack dominated by 0. These results will play a crucial role in the next sections. First, we consider some preliminaries on noncommutative Poisson transforms. Let C^{∗}(S_{1}, …, S_{n}) be the Cuntz-Toeplitz C^{∗}-algebra generated by the left creation operators (see [11]). The noncommutative Poisson transform at the ntuple T := (T_{1}, …, T_{n}) ∈ [B(H)^{n}]_{1}^{−} is the unital completely contractive linear map P_{T} : C^{∗}(S_{1}, …, S_{n}) → B(H) defined by

P_{T} [f] := lim_{r→1} K_{rT}^{∗} (I_{H} ⊗ f)K_{rT} , f ∈ C^{∗}(S_{1}, …, S_{n}),

where the limit exists in the operator norm topology of B(H). Here, the noncommutative Poisson kernel K_{rT} : H → \overline{Δ_{rT} H} ⊗ F^{2}(H_{n}), 0 < r ≤ 1, is defined by

K_{rT} h := ∑_{k=0}^{∞} ∑_{|α|=k} r^{|α|}Δ_{rT} T_{α}^{∗}h ⊗ e_{α}, h ∈ H,

where {e_{α}}_{α∈𝔽_{n}^{+}} is the orthonormal basis for the full Fock space F^{2}(H_{n}), defined by e_{α} := e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{k}} if α = g_{i_{1}} ⋯ g_{i_{k}} ∈ 𝔽_{n}^{+} and e_{g_{0}} := 1, and Δ_{rT} := (I_{H} − r^{2}T_{1}T_{1}^{∗} −⋯−r^{2}T_{n}T_{n}^{∗})^{1/2}. We recall that P_{T} [S_{α}S_{β}^{∗}] = T_{α}T_{β}^{∗}, α, β ∈ 𝔽_{n}^{+}. When T := (T_{1}, …, T_{n}) is a pure row contraction, i.e., SOT- lim_{k→∞} ∑_{|α|=k} T_{α}T_{α}^{∗} = 0, then we have

P_{T} [f] = K_{T}^{∗} (I_{D_{T}} ⊗ f)K_{T} , f ∈ C^{∗}(S_{1}, …, S_{n}) or f ∈ F_{n}^{∞},

where D_{T} := \overline{Δ_{T} H}. We refer to [41], [42], and [48] for more on noncommutative Poisson transforms on C^{∗}-algebras generated by isometries. A free pluriharmonic function u on [B(K)^{n}]_{1} with operator valued coefficients is called positive, and denote u ≥ 0, if u(X_{1}, …, X_{n}) ≥ 0 for any (X_{1}, …, X_{n}) ∈ [B(K)^{n}]_{1}, where K is an infinite dimensional Hilbert space. We mention that it is enough to assume that the positivity condition holds for any finite dimensional Hilbert space K. Indeed, for each m ∈ ℕ, consider R^{(m)} := (R_{1}^{(m)}, …, R_{n}^{(m)}), where R_{i}^{(m)} is the compression of the right creation operator R_{i} to the subspace P_{m} := span {e_{α} : α ∈ 𝔽_{n}^{+}, |α| ≤ m} of F^{2}(H_{n}). We recall from [47] the following result.

Lemma 2.1. Let u be a free pluriharmonic function on [B(K)^{n}]_{1} with operatorvalued coefficients. Then u(X_{1}, …, X_{n}) ≥ 0 for any (X_{1}, …, X_{n}) ∈ [B(K)^{n}]_{1} if and only if u(R_{1}^{(m)}, …, R_{n}^{(m)}) ≥ 0 for any m ∈ ℕ.

Let A := (A_{1}, …, A_{n}) and B := (B_{1}, …, B_{n}) be n-tuples of operators in

C_{ρ} ⊂ B(H)^{n}. We say that A is Harnack dominated by B, and denote A ≺^{H} B, if there exists c > 0 such that

ℜp(A_{1}, …, A_{n}) + (ρ − 1)ℜp(0) ≤ c^{2} [ℜp(B_{1}, …, B_{n}) + (ρ − 1)ℜp(0)]

for any noncommutative polynomial with matrix-valued coefficients p ∈ ℂ[X_{1}, …, X_{n}] ⊗ M_{m}, m ∈ ℕ, such that ℜp ≥ 0. When we want to emphasize

the constant c, we write A≺_{c}^{H} B.

According to Theorem 1.3, we can associate with each n-tuple T := (T_{1}, …, T_{n}) ∈ C_{ρ} the completely positive map φ_{T} : \overline{A_{n}^{∗} + A_{n}}^{∥·∥} → B(H) defined by

(2.1)  φ_{T} (g) := \frac{1}{ρ}g(T_{1}, …, T_{n}) + (1 − \frac{1}{ρ}) g(0),

where g(T_{1}, …, T_{n}) is defined by the free pluriharmonic functional calculus for the class C_{ρ}. Now, we present several characterizations for the Harnack domination in C_{ρ}.

Theorem 2.2. Let A := (A_{1}, …, A_{n}) ∈ B(H)^{n} and B := (B_{1}, …, B_{n}) ∈ B(H)^{n} be in the class C_{ρ} and let c > 0. Then the following statements are equivalent:

(i) A≺_{c}^{H} B;

(ii) P_{ρ}(rA, R) ≤ c^{2}P_{ρ}(rB, R) for any r ∈ [0, 1), where P_{ρ}(X, R) is the

multi-Toeplitz kernel associated with X ∈ C_{ρ}; (iii) u(rA_{1}, …, rA_{n}) + (ρ − 1)u(0) ≤ c^{2} [u(rB_{1}, …, rB_{n}) + (ρ − 1)u(0)] for

any positive free pluriharmonic function u on [B(H)^{n}]_{1} with operatorvalued coefficients and any r ∈ [0, 1);

(iv) K_{ρ,A} ≤ c^{2}K_{ρ,B}, where K_{ρ,X} is the multi-Toeplitz kernel associated with

X ∈ C_{ρ};

(v) c^{2}φ_{B} − φ_{A} is a completely positive linear map on the operator space \overline{A_{n}^{∗} + A_{n}}^{∥·∥}, where φ_{A}, φ_{B} are the c.p. maps associated with A and B, respectively.

(vi) there is an operator L_{B,A} ∈ B(K_{B}, K_{A}) with ∥L_{B,A}∥ ≤ c such that L_{B,A}|_{H} = I_{H} and

L_{B,A}W_{i} = V_{i}L_{B,A}, i = 1, …, n,

where (V_{1}, …, V_{n}) on K_{A} ⊃ H and (W_{1}, …, W_{n}) on K_{A} ⊃ H are the minimal isometric dilations of A and B, respectively.

Proof. First we prove that (i) ⟹ (ii). Since R_{α}^{(m)} = 0 for any α ∈ 𝔽_{n}^{+} with |α| ≥ m + 1, we have

P_{ρ}(rX, R^{(m)}) = ∑_{1≤|α|≤m} r^{|α|}X_{α}^{∗} ⊗ R(m) ; α̃ + ρI ⊗ I + ∑_{1≤|α|≤m} r^{|α|}X_{α} ⊗ R(m) ; α̃^{∗}.

Since X ↦ P_{1}(X, R) is a positive free pluriharmonic function on [B(H)^{n}]_{1}, with coefficients in B(F^{2}(H_{n})), so is the map

X ↦ P_{1}(rX, R^{(m)}) = (I ⊗ P_{P_{m}})P_{1}(rX, R)|_{H⊗P_{m}} for any r ∈ [0, 1). If A≺_{c}^{H} B, then we have

P_{1}(rA, R^{(m)}) + (ρ − 1)P_{1}(0, R^{(m)}) ≤ c^{2} [P_{1}(rB, R^{(m)}) + (ρ − 1)P_{1}(0, R^{(m)})]

for any m = 1, 2, …. Using Lemma 2.1, we deduce that

P_{1}(rA, R) + (ρ − 1)I ≤ c^{2} [P_{1}(rB, R) + (ρ − 1)I]

for any r ∈ [0, 1). Since P_{ρ}(rY, R) = P_{1}(rY, R) + (ρ − 1)I for any n-tuple Y ∈ B(H)^{n} with spectral radius r(Y) ≤ 1 and r ∈ [0, 1), we deduce item (ii). To prove the implication (ii) ⟹ (iii), assume that condition (ii) holds and let u be a positive free pluriharmonic function on [B(H)^{n}]_{1} with coefficients in B(E) of the form

u(Z_{1}, …, Z_{n}) = ∑_{k=1}^{∞} ∑_{|α|=k} Z_{α}^{∗} ⊗ C_{(α)}^{∗} + I ⊗ C_{(0)} + ∑_{k=1}^{∞} ∑_{|α|=k} Z_{α} ⊗ C_{(α)}.

It is well-known (see e.g. [29]) that if S ⊆ B(F^{2}(H_{n})) is an operator system and μ : S → B(K) is a completely bounded map, then there exists a completely bounded linear map

μ̃ := μ ⊗ id : S ¯⊗_{min}B(H) → B(K) ¯⊗_{min}B(H)

such that μ̃(f ⊗ Y) := μ(f) ⊗ Y for f ∈ S and Y ∈ B(H). Moreover, ∥μ̃∥_{cb} = ∥μ∥_{cb} and, if μ is completely positive, then so is μ̃. Using Corollary 5.5 from [47], we find a completely positive linear map ν : R_{n}^{∗} + R_{n} → B(E) such that ν(Rα̃) = C_{(α)}^{∗} if |α| ≥ 1 and ν(I) = C_{(0)}. Note that

(id ⊗ ν)[c^{2}P_{ρ}(rB, R) − P_{ρ}(rA, R)]

= (id ⊗ ν) {∑_{k=1}^{∞} ∑_{|α|=k} r^{|α|}(c^{2}B_{α} − A_{α}) ⊗ R∗ ; α̃ + ρ(c^{2} − 1)I ⊗ I

+ ∑_{k=1}^{∞} ∑_{|α|=k}(c^{2}B_{α}^{∗} − A_{α}^{∗}) ⊗ Rα̃}

= {∑_{k=1}^{∞} ∑_{|α|=k} r^{|α|}(c^{2}B_{α} − A_{α}) ⊗ C_{(α)} + ρ(c^{2} − 1)I ⊗ C_{(0)}

+ ∑_{k=1}^{∞} ∑_{|α|=k}(c^{2}B_{α}^{∗} − A_{α}^{∗}) ⊗ C_{(α)}^{∗}}

= c^{2} [u(rB_{1}, …, rB_{n}) + (ρ − 1)u(0)]

−[u(rA_{1}, …, rA_{n}) + (ρ − 1)u(0)] .

Hence, and using the fact that c^{2}P_{ρ}(rB, R) − P_{ρ}(rA, R) ≥ 0, we deduce that

c^{2} [u(rB_{1}, …, rB_{n}) + (ρ − 1)u(0)] − [u(rA_{1}, …, rA_{n}) + (ρ − 1)u(0)] ≥ 0, which proves (iii).

Now, we prove the implication (iii) ⟹ (v). Let g ∈ (\overline{A_{n}^{∗} + A_{n}}^{∥·∥}) ⊗_{min} M_{m} be positive. Then, according to Theorem 4.1 from [47], the map defined by

g(X) := (P_{X} ⊗ id)[g], X ∈ [B(H)^{n}]_{1},

is a positive free pluriharmonic function. Condition (iii) implies

g(rA_{1}, …, rA_{n}) + (ρ − 1)g(0) ≤ c^{2} [g(rB_{1}, …, rB_{n}) + (ρ − 1)g(0)]

for any r ∈ [0, 1). Hence, and using relation (2.1), we get ρφ_{A}(g_{r}) ≤ c^{2}ρφ_{B}(g_{r}). Taking r → 1, we deduce item (v). To prove the implication (v) ⟹ (i), let p ∈ ℂ[X_{1}, …, X_{n}] ⊗ M_{m}, m ∈ ℕ, be a noncommutative polynomial with matrix coefficients such that Re p ≥ 0. Since

ρφ_{Y} (p) = p(Y_{1}, …, Y_{n}) + (ρ − 1)p(0)

for any Y := (Y_{1}, …, Y_{n}) ∈ C_{ρ}, it is clear that (v) implies item (i). We prove now that (ii) ⟹ (iv). We recall that e_{α} := e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{k}} if α = g_{i_{1}} ⋯ g_{i_{k}} ∈ 𝔽_{n}^{+} and e_{g_{0}} := 1, and that {e_{α}}_{α∈𝔽_{n}^{+}} is an orthonormal basis for the full Fock space F^{2}(H_{n}). First, we prove that (2.2)

⟨P_{ρ}(X, rR) ( ∑_{|β|≤q} h_{β} ⊗ e_{β}) , ∑_{|γ|≤q} h_{γ} ⊗ e_{γ}⟩ = ρ ∑_{|β|,|γ|≤q} ⟨K_{ρ,X,r}(γ, β)h_{β}, h_{γ}⟩ ,

where the multi-Toeplitz kernel K_{ρ,X,r} : 𝔽_{n}^{+} ×𝔽_{n}^{+} → B(H), r ∈ (0, 1), is defined by

K_{ρ,X,r}(α, β) := \begin{cases}\frac{1}{ρ}r^{|β\_{l}α|}X_{β\_{l}α} & if β >_{l} α \\ I & if α = β \\ \frac{1}{ρ}r^{|α\_{l}β|}(X_{α\_{l}β})^{∗} & if α >_{l} β \\ 0 & otherwise. \end{cases}

Note that if {h_{β}}_{|β|≤q} ⊂ H, then we have

⟨(ρI ⊗ I + ∑_{k=1}^{∞} ∑_{|α|=k} X_{α}^{∗} ⊗ r^{k}Rα̃) ( ∑_{|β|≤q} h_{β} ⊗ e_{β}) , ∑_{|γ|≤q} h_{γ} ⊗ e_{γ}⟩

= ρ ∑_{|β|≤q} ∥h_{β}∥^{2} + ∑_{k=1}^{∞} ∑_{|α|=k} ⟨ ∑_{|β|≤q} X_{α}^{∗}h_{β} ⊗ r^{k}Rα̃e_{β}, ∑_{|γ|≤q} h_{γ} ⊗ e_{γ}⟩

= ρ ∑_{|β|≤q} ∥h_{β}∥^{2} + + ∑_{|α|≥1} ∑_{|β|,|γ|≤q} r^{|α|} ⟨e_{βα}, e_{γ}⟩ ⟨X_{α}^{∗}h_{β}, h_{γ}⟩

= ρ ∑_{|β|≤q} ∥h_{β}∥^{2} + ∑_{γ>β; |β|,|γ|≤q} r^{|γ\_{l}β|} ⟨X_{γ\_{l}β}^{∗}h_{β}, h_{γ}⟩

= ∑_{γ≥β; |β|,|γ|≤q} ⟨ρK_{ρ,X,r}(γ, β)h_{β}, h_{γ}⟩ .

Now, taking into account that K_{ρ,X,r}(γ, β) = K_{ρ,X,r}^{∗}(β, γ), we deduce relation (2.2). Therefore, the condition P_{ρ}(rA, R) ≤ c^{2}P_{ρ}(rB, R), r ∈ [0, 1), implies

[K_{ρ,A,r}(α, β)]_{|α|,|β|≤q} ≤ c^{2}[K_{ρ,B,r}(α, β)]_{|α|,|β|≤q}

for any 0 < r < 1 and q = 0, 1, . . .. Taking r → 1 in the latter inequality, we obtain item (iv). Assume now that (iv) holds. Since c^{2}K_{ρ,B} − K_{ρ,A} is a positive semidefinite multi-Toeplitz kernel, due to Theorem 3.1 from [39] (see also the proof of Theorem 5.2 from [47]), we find a completely positive linear map μ : C^{∗}(S_{1}, …, S_{n}) → B(E) such that

μ(S_{α}) = c^{2}K_{ρ,B}(g_{0}, α) − K_{ρ,A}(g_{0}, α) = \frac{1}{ρ}(c^{2}B_{α} − A_{α})

for any α ∈ 𝔽_{n}^{+} with |α| ≥ 1, and μ(I) = (c^{2} − 1)I. Since

P (rS, R) := ∑_{k=1}^{∞} ∑_{|α|=k} r^{k}S_{α} ⊗ R∗ ; α̃ + I ⊗ I + ∑_{k=1}^{∞} ∑_{|α|=k} r^{k}S_{α}^{∗} ⊗ Rα̃ ≥ 0

for r ∈ [0, 1), we deduce that

(μ ⊗ id)[P (rS, R)] = ∑_{k=1}^{∞} ∑_{|α|=k} \frac{1}{ρ}r^{|α|}[c^{2}B_{α}^{∗} − A_{α}^{∗}] ⊗ Rα̃ + (c^{2} − 1)I ⊗ I

+ ∑_{k=1}^{∞} ∑_{|α|=k} \frac{1}{ρ}r^{|α|}[c^{2}B_{α} − A_{α}] ⊗ R∗ ; α̃

= c^{2}P_{ρ}(rB, R) − P_{ρ}(rA, R) ≥ 0,

which implies (ii). Let us prove that (iv) ⟹ (vi). Assume that (iv) holds. Then we have K_{ρ,A} ≤ c^{2}K_{ρ,B}, where K_{ρ,X} is the multi-Toeplitz kernel associated with X ∈ C_{ρ}. Let V := (V_{1}, …, V_{n}) be the minimal isometric dilation of A := (A_{1}, …, A_{n}). Then K_{A} = ⋁_{α∈𝔽_{n}^{+}} V_{α}H and ρP_{H}V_{α}|_{H} = A_{α} for any |α| ≥ 1. Similar properties hold if W := (W_{1}, …, W_{n}) is the minimal isometric dilation of B := (B_{1}, …, B_{n}). Hence, and taking into account that V_{1}, …, V_{n} and W_{1}, …, W_{n} are isometries with orthogonal ranges, respectively, we have

∥ ∑_{|α|≤m} V_{α}h_{α}∥^{2} =

= ∑_{α>_{l}β,|α|,|β|≤m} ⟨V_{α\_{l}β}h_{α}, h_{β}⟩ + ∑_{|α|≤m} ⟨h_{α}, h_{α}⟩ + ∑_{β>_{l}α,|α|,|β|≤m} ⟨V_{β\_{l}α}^{∗}h_{α}, h_{β}⟩

= ∑_{α>_{l}β,|α|,|β|≤m} ⟨ \frac{1}{ρ} A_{α\_{l}β}h_{α}, h_{β}⟩ + ∑_{|α|≤m} ⟨h_{α}, h_{α}⟩ + ∑_{β>_{l}α,|α|,|β|≤m} ⟨ \frac{1}{ρ} A_{β\_{l}α}^{∗}h_{α}, h_{β}⟩

= ∑_{|α|≤m,|β|≤m} ⟨K_{ρ,A}(β, α)h_{α}, h_{β}⟩ = ⟨[K_{ρ,A}(β, α)]_{|α|,|β|≤m} h_{m}, h_{m}⟩

for any m ∈ ℕ and h_{m} := ⊕_{|α|≤m}h_{α} ∈ ⊕_{|α|≤m}H_{α}, where each H_{α} is a copy of H. Similarly, we obtain

∥ ∑_{|α|≤m} W_{α}h_{α}∥^{2} = ⟨[K_{ρ,B}(β, α)]_{|α|,|β|≤m} h_{m}, h_{m}⟩ .

Taking into account that K_{ρ,A} ≤ c^{2}K_{ρ,B}, we deduce that

∥ ∑_{|α|≤m} V_{α}h_{α}∥ ≤ c ∥ ∑_{|α|≤m} W_{α}h_{α}∥ .

Therefore, we can define an operator L_{B,A} : K_{B} → K_{A} by setting

(2.3)  L_{B,A} ( ∑_{|α|≤m} W_{α}h_{α}) := ∑_{|α|≤m} V_{α}h_{α}

for any m ∈ ℕ and h_{α} ∈ H, α ∈ 𝔽_{n}^{+}. Note that L_{B,A} is a bounded operator with ∥L_{B,A}∥ ≤ c. Since L_{B,A}|_{H} = I_{H}, we have ∥L_{B,A}∥ ≥ 1. It is easy to see that L_{B,A}W_{i} = V_{i}L_{B,A} for i = 1, …, n. Therefore item (vi) holds.

Conversely, assume that there is an operator L_{B,A} ∈ B(K_{B}, K_{A}) with norm ∥L_{B,A}∥ ≤ c such that L_{B,A}|_{H} = I_{H} and L_{B,A}W_{i} = V_{i}L_{B,A}, i = 1, …, n. Then, we deduce that L_{B,A} (∑_{|α|≤m} W_{α}h_{α}) = ∑_{|α|≤m} V_{α}h_{α} for any m ∈ ℕ and h_{α} ∈ H, α ∈ 𝔽_{n}^{+}. The condition ∥L_{B,A}∥ ≤ c implies

∥ ∑_{|α|≤m} V_{α}h_{α}∥^{2} ≤ c^{2} ∥ ∑_{|α|≤m} W_{α}h_{α}∥^{2} ,

which is equivalent to the inequality

⟨[K_{ρ,A}(β, α)]_{|α|,|β|≤m} h_{m}, h_{m}⟩ ≤ c^{2} ⟨[K_{ρ,B}(β, α)]_{|α|,|β|≤m} h_{m}, h_{m}⟩

for any m ∈ ℕ and h_{m} := ⊕_{|α|≤m}h_{α} ∈ ⊕_{|α|≤m}H_{α}. Consequently, we deduce item (iv). The proof is complete. □

A closer look at the proof of Theorem 2.2 reveals that one can assume that u(0) = I in part (iii), and one can also assume that ℜp(0) = I in the definition

of the Harnack domination A≺^{H} B. We also remark that, due to Theorem 1.3,

we can add an equivalence to Theorem 2.2, namely, A≺_{c}^{H} B if and only if

u(A_{1}, …, A_{n}) + (ρ − 1)u(0) ≤ c^{2} [u(B_{1}, …, B_{n}) + (ρ − 1)u(0)]

for any positive free pluriharmonic function u ∈ Har_{ball}^{c}(B(E)).

Corollary 2.3. If A, B ∈ C_{ρ} and A≺^{H} B, then

∥L_{B,A}∥ = inf{c > 1 : A≺_{c}^{H} B}

= inf{c > 1 : P_{ρ}(rA, R) ≤ c^{2}P_{ρ}(rB, R) for any r ∈ [0, 1)}.

Moreover, A≺^{H} B if and only if sup_{r∈[0,1)} ∥L_{rA,rB}∥ < ∞. In this case,

∥L_{A,B}∥ = sup_{r∈[0,1)} ∥L_{rA,rB}∥

and the mapping r ↦ ∥L_{rA,rB}∥ is increasing on [0, 1).

Proof. Assume that A≺^{H} B. Then, due to Theorem 2.2, A≺_{c}^{H} B if and only if there

is an operator L_{B,A} ∈ B(K_{B}, K_{A}) with ∥L_{B,A}∥ ≤ c such that L_{B,A}|_{H} = I_{H} and L_{B,A}W_{i} = V_{i}L_{B,A} for i = 1, …, n. Consequently, taking c = ∥L_{B,A}∥, we

deduce that A ≺_{∥L_{B,A}∥}^{H} B, which is equivalent to

P_{ρ}(rA, R) ≤ ∥L_{B,A}∥^{2}P_{ρ}(rB, R)

for any r ∈ [0, 1). Hence, we have tA ≺_{∥L_{B,A}∥}^{H} tB for any t ∈ [0, 1). Applying again

Theorem 2.2 to the operators tA and tB, we deduce that ∥L_{tA,tB}∥ ≤ ∥L_{B,A}∥. Conversely, suppose that c := sup_{r∈[0,1)} ∥L_{rA,rB}∥ < ∞. Since ∥L_{rA,rB}∥ ≤ c,

Theorem 2.2 implies rA≺_{c}^{H} rB for any r ∈ [0, 1) and, therefore, P_{ρ}(rtA, R) ≤

c^{2}P_{ρ}(rtB, R) for any t, r ∈ [0, 1). Hence, A≺_{c}^{H} B and, consequently, ∥L_{B,A}∥ ≤ c.

Therefore, ∥L_{A,B}∥ = sup_{r∈[0,1)} ∥L_{rA,rB}∥. The fact that r ↦ ∥L_{rA,rB}∥ is an increasing function on [0, 1) follows from the latter relation. This completes the proof. □

We remark that if 1 ≤ m < n and u is a positive free pluriharmonic function on [B(K)^{n}]_{1}, then the map

(X_{1}, …, X_{m}) ↦ u(X_{1}, …, X_{m}, 0, …, 0)

is a positive free pluriharmonic function on [B(K)^{m}]_{1}. Moreover, if g is a positive free pluriharmonic function on [B(K)^{m}]_{1}, then the map

(X_{1}, …, X_{n}) ↦ g(X_{1}, …, X_{m}, 0, …, 0)

is a positive free pluriharmonic function on [B(K)^{n}]_{1}. Consequently, using Corollary 1.2, one can easily deduce the following result.

Corollary 2.4. Let c > 0, ρ > 0, and 1 ≤ m < n. Consider two n-tuples (A_{1}, …, A_{m}) ∈ B(H)^{m} and (B_{1}, …, B_{m}) ∈ B(H)^{m} in the class C_{ρ} and let (A_{1}, …, A_{m}, 0, …, 0) and (B_{1}, …, B_{m}, 0, …, 0) be their extensions in B(H)^{n},

respectively. Then (A_{1}, …, A_{m})≺_{c}^{H} (B_{1}, …, B_{m}) in C_{ρ} ⊂ B(H)^{m} if and only if

(A_{1}, …, A_{m}, 0, …, 0)≺_{c}^{H} (B_{1}, …, B_{m}, 0, …, 0) in C_{ρ} ⊂ B(H)^{n}.

We recall (e.g. [43]) that if (T_{1}, … T_{n}) is an n-tuple of operators, then the joint spectral radius r(T_{1}, …, T_{n}) < 1 if and only if lim_{k→∞} ∥∑_{|α|=k} T_{α}T_{α}^{∗}∥ = 0.

In what follows, we characterize the elements of C_{ρ} which are Harnack dominated by 0.

Theorem 2.5. Let A := (A_{1}, …, A_{n}) be in C_{ρ}. Then A≺^{H} 0 if and only if the joint spectral radius r(A_{1}, …, A_{n}) < 1.

Proof. Note that the map X ↦ P_{ρ}(X, R) is a positive free pluriharmonic function on [B(H)^{n}]_{1} with coefficients in B(F^{2}(H_{n})) and has the factorization

\begin{matrix}P_{ρ}(X, R) = \\ = (I − R_{X})^{−1} + (ρ − 2)I + (I − R_{X}^{∗})^{−1} \\ = (I − R_{X}^{∗})^{−1} [I − R_{X} + (ρ − 2)(I − R_{X}^{∗})(I − R_{X}) + I − R_{X}^{∗}] (I − R_{X})^{−1} \\ = (I − R_{X}^{∗})^{−1} [ρI + (1 − ρ)(R_{X}^{∗} + R_{X}) + (ρ − 2)R_{X}^{∗}R_{X}] (I − R_{X})^{−1}, \end{matrix} (2.4)

where R_{X} := X_{1}^{∗} ⊗ R_{1} + ⋯ + X_{n}^{∗} ⊗ R_{n} is the reconstruction operator associated with the n-tuple X := (X_{1}, …, X_{n}) ∈ [B(H)^{n}]_{1}. We remark that, due to the fact that the spectral radius of R_{X} is equal to the joint spectral radius r(X_{1}, …, X_{n}), the factorization above holds for any X ∈ C_{ρ} with r(X_{1}, …, X_{n}) < 1. Now, using Theorem 2.2 part (ii) and the above-mentioned factorization, we

deduce that A≺^{H} 0 if and only if there exists c > 0 such that

(I − R_{rA}^{∗})^{−1} [ρI + (1 − ρ)(R_{rA}^{∗} + R_{rA}) + (ρ − 2)R_{rA}^{∗}R_{rA}] (I − R_{rA})^{−1} ≤ ρc^{2}I

for any r ∈ [0, 1). Similar inequality holds if we replace the right creation operators by the left creation operators. Then, applying the noncommutative Poisson transform id ⊗ P_{e^{iθ}R}, where R := (R_{1}, …, R_{n}), we obtain (2.5) ρI +(1−ρ)(e^{−iθ}R_{rA}^{∗}+e^{iθ}R_{rA})+(ρ−2)R_{rA}^{∗}R_{rA} ≤ ρc^{2}(I −re^{−iθ}R_{A}^{∗})(I −re^{iθ}R_{A})

for any r ∈ [0, 1) and θ ∈ ℝ. On the other hand, since A := (A_{1}, …, A_{n}) ∈ C_{ρ}, we have r(A_{1}, …, A_{n}) ≤ 1. Suppose that r(A_{1}, …, A_{n}) = 1. Taking into account that r(R_{A}) = r(A_{1}, …, A_{n}), we can find λ_{0} ∈ 𝕋 in the approximative spectrum of R_{A}. Consequently, there is a sequence {h_{m}} in H ⊗ F^{2}(H_{n}) such that ∥h_{m}∥ = 1 and

(2.6)  λ_{0}h_{m} − R_{A}h_{m} → 0 as m → ∞.

In particular, relation (2.5) implies

\begin{aligned}ρ∥h_{m}∥^{2} & + (1 − ρ) [⟨λ_{0}R_{rA}^{∗}h_{m}, h_{m}⟩ + ⟨λ̄_{0}R_{rA}h_{m}, h_{m}⟩] + (ρ − 2)∥R_{rA}h_{m}∥^{2} \\ & ≤ ρc^{2}∥h_{m} − λ̄_{0}R_{rA}h_{m}∥^{2} \end{aligned} (2.7)

for any r ∈ (0, 1) and m ∈ ℕ. Note that due to (2.6) and the fact that |λ_{0}| = 1, we have

⟨λ̄_{0}R_{A}h_{m}, h_{m}⟩ = λ̄_{0} ⟨R_{A}h_{m} − λ_{0}h_{m}, h_{m}⟩ + 1 → 1, as m → ∞.

Since

\begin{aligned}∥h_{m} − λ̄_{0}R_{rA}h_{m}∥ & ≤ ∥h_{m} − λ̄_{0}R_{A}h_{m}∥ + ∥λ̄_{0}(R_{A}h_{m} − R_{rA}h_{m})∥ \\ & = ∥λ̄_{0}h_{m} − R_{A}h_{m}∥ + (1 − r)∥R_{A}h_{m}∥ \end{aligned}

and due to the fact that ∥R_{A}h_{m}∥ → 1 as m → ∞, we deduce that

lim sup_{m→∞} ∥h_{m} − λ̄_{0}R_{rA}h_{m}∥ ≤ 1 − r

for any r ∈ (0, 1). Now, since R_{rA} = rR_{A} and taking m → ∞ in relation (2.7), we obtain

ρ + 2(1 − ρ)r + (ρ − 2)r^{2} ≤ c^{2}ρ(1 − r)^{2}

for any r ∈ (0, 1). Setting r = 1 − \frac{1}{m}, m ≥ 2, straightforward calculations imply 2m ≤ ρc^{2} − ρ + 2 for any m ∈ ℕ, which is a contradiction. Therefore, we must have r(A_{1}, …, A_{n}) < 1. Conversely, assume that A := (A_{1}, …, A_{n}) ∈ C_{ρ} has the joint spectral radius r(A_{1}, …, A_{n}) < 1. Since r(A_{1}, …, A_{n}) = r(R_{A}), one can see that M := sup_{r∈(0,1)} ∥(I − rR_{A})^{−1}∥ exists and M ≥ 1. Hence

(2.8)  M^{2}(I − R_{rA}^{∗})(I − R_{rA}) ≥ I ≥ I − R_{rA}^{∗}R_{rA}

for any r ∈ (0, 1). Now we consider the case ρ ≥ 1. Note that relation (2.8) implies

I − R_{rA}^{∗}R_{rA} + (ρ − 1)(I − R_{rA}^{∗})(I − R_{rA}) ≤ ρM^{2}(I − R_{rA}^{∗})(I − R_{rA}).

The latter inequality is equivalent to

ρI + (1 − ρ)(R_{rA}^{∗} + R_{rA}) + (ρ − 2)R_{rA}^{∗}R_{rA} ≤ ρM^{2}(I − R_{rA}^{∗})(I − R_{rA}),

which, due to the factorization (2.4), is equivalent to

P_{ρ}(rA, R) ≤ ρM^{2} = M^{2}P_{ρ}(0, R)

for any r ∈ [0, 1). According to Theorem 2.2, we deduce that A≺^{H} 0. Now, consider the case when ρ ∈ (0, 1). Since ∥R_{rA}∥ ≤ rρ and δ − 2 < 0, we have

\begin{aligned}ρI + (1 − ρ)(R_{rA}^{∗} + R_{rA}) + (ρ − 2)R_{rA}^{∗}R_{rA} & ≤ ρI + (1 − ρ)(R_{rA}^{∗} + R_{rA}) \\ & ≤ ρI + 2(1 − ρ)rρ ≤ (3ρ − 2ρ^{2})I. \end{aligned}

Using again the factorization (2.4), we deduce that

P_{ρ}(rA, R) ≤ (3ρ − 2ρ^{2})(I − R_{rA}^{∗})^{−1}(I − R_{rA})^{−1} for any r ∈ (0, 1). Hence and using the fact that (I−R_{rA}^{∗})^{−1}(I−R_{rA})^{−1} ≤ M^{2}I, we obtain

P_{ρ}(rA, R) ≤ (3 − 2ρ)M^{2}P_{ρ}(0, R)

for any r ∈ (0, 1). Using again Theorem 2.2, we get A≺^{H} 0. The proof is complete. □

We mention that in the particular case when n = 1 we can recover a result obtained by Ando, Suciu, and Timotin [1], when ρ = 1, and by G. Cassier and N. Suciu [9], when ρ ≠ 1.

3. Hyperbolic metric on Harnack parts of the noncommutative ball C_{ρ}

The relation ≺^{H} induces an equivalence relation ∼^{H} on the class C_{ρ}. We provide a Harnack type double inequality for positive free pluriharmonic functions on the noncommutative ball C_{ρ} and use it to prove that the Harnack part of C_{ρ} which contains 0 coincides with the open noncommutative ball [C_{ρ}]_{<1}. We introduce a hyperbolic metric on any Harnack part of C_{ρ} and obtain a concrete formula in terms of the reconstruction operator.

Since ≺^{H} is a preorder relation on C_{ρ}, it induces an equivalence relation ∼^{H} on C_{ρ}, which we call Harnack equivalence. The equivalence classes with respect to ∼^{H} are called Harnack parts of C_{ρ}. Let A := (A_{1}, …, A_{n}) and B := (B_{1}, …, B_{n}) be in C_{ρ}. We say that A and B are Harnack equivalent (we denote A ∼^{H} B) if and only if there exists c ≥ 1 such that

\begin{aligned}\frac{1}{c^{2}} [ℜp(B_{1}, …, B_{n}) + (ρ − 1)ℜp(0)] & ≤ ℜp(A_{1}, …, A_{n}) + (ρ − 1)ℜp(0) \\ & ≤ c^{2} [ℜp(B_{1}, …, B_{n}) + (ρ − 1)ℜp(0)] \end{aligned}

for any noncommutative polynomial with matrix-valued coefficients p ∈ ℂ[X_{1}, …, X_{n}] ⊗ M_{m}, m ∈ ℕ, such that ℜp(X) ≥ 0 for any X ∈ [B(H)^{n}]_{1}.

We also use the notation A∼_{c}^{H} B when A≺_{c}^{H} B and B≺_{c}^{H} A. We remark that Theorem 2.2 can be used to provide several characterizations for the Harnack parts of C_{ρ}. The first result is an extension of Harnack inequality to positive free pluriharmonic functions on the noncommutative ball C_{ρ}, ρ > 0.

Theorem 3.1. If u is a positive free pluriharmonic function on [B(H)^{n}]_{1} with operator-valued coefficients in B(E) and 0 ≤ r < 1, then

u(0) \frac{1 − r(2ρ − 1)}{1 + r} ≤ u(rX_{1}, …, rX_{n}) ≤ u(0) \frac{1 + r(2ρ − 1)}{1 − r}

for any (X_{1}, …, X_{n}) ∈ C_{ρ}.

Proof. Let

u(Z_{1}, …, Z_{n}) = ∑_{k=1}^{∞} ∑_{|α|=k} Z_{α}^{∗} ⊗ A_{(α)}^{∗} + I ⊗ A_{(0)} + ∑_{k=1}^{∞} ∑_{|α|=k} Z_{α} ⊗ A_{(α)}

be a positive free pluriharmonic function on [B(H)^{n}]_{1} with coefficients in B(E). According to Theorem 1.4 from [49], for any Y ∈ [B(H)^{n}]_{1}^{−} and r ∈ [0, 1), we have

(3.1)  u(0) \frac{1 − r}{1 + r} ≤ u(rY_{1}, …, rY_{n}) ≤ u(0) \frac{1 + r}{1 − r} .

On the other hand, let (X_{1}, …, X_{n}) ∈ C_{ρ} and let (V_{1}, …, V_{n}) be the minimal isometric dilation of (X_{1}, …, X_{n}) on a Hilbert space K_{T} ⊇ H. Since X_{α} = ρP_{H}V_{α}|_{H} for any α ∈ 𝔽_{n}^{+}\{g_{0}}, and using the free pluriharmonic functional calculus, we have

u(rX_{1}, …, rX_{n}) =

= ∑_{k=1}^{∞} ∑_{|α|=k} r^{|α|}X_{α}^{∗} ⊗ A_{(α)}^{∗} + I ⊗ A_{(0)} + ∑_{k=1}^{∞} ∑_{|α|=k} r^{|α|}X_{α} ⊗ A_{(α)}

= ρ(P_{H} ⊗ I_{E}) [∑_{k=1}^{∞} ∑_{|α|=k} r^{|α|}V_{α}^{∗} ⊗ A_{(α)}^{∗}] |_{H⊗E} + I_{H} ⊗ A_{(0)}

+ ρ(P_{H} ⊗ I_{E}) [∑_{k=1}^{∞} ∑_{|α|=k} r^{|α|}V_{α} ⊗ A_{(α)}] |_{H⊗E}

= ρ(P_{H} ⊗ I_{E})u(rV_{1}, …, rV_{n})|_{H⊗E} + (1 − ρ)u(0),

where the convergence is in the operator norm topology. Due to (3.1), we have

u(0) \frac{1 − r}{1 + r} ≤ u(rV_{1}, …, V_{n}) ≤ u(0) \frac{1 + r}{1 − r} .

Consequently, we deduce that

u(0) [\frac{ρ(1 − r)}{1 + r} + (1 − ρ)] ≤ ρ(P_{H} ⊗ I_{E})u(rV_{1}, …, rV_{n})|_{H⊗E} + (1 − ρ)u(0)

≤ u(0) [\frac{ρ(1 + r)}{1 − r} + (1 − ρ)] .

Since

u(rX_{1}, …, rX_{n}) = ρ(P_{H} ⊗ I_{E})u(rV_{1}, …, rV_{n})|_{H⊗E} + (1 − ρ)u(0),

the result follows. □

Now, we can determine the Harnack part of C_{ρ} which contains 0.

Theorem 3.2. Let A := (A_{1}, …, A_{n}) be in C_{ρ}. Then the following statements are equivalent:

(i) ω_{ρ}(A_{1}, …, A_{n}) < 1;

(ii) A ∼^{H} 0;

(iii) r(A_{1}, …, A_{n}) < 1 and P_{ρ}(A, R) ≥ aI for some constant a > 0.

Proof. First, we prove that (i) ⟹ (ii). Let A := (A_{1}, …, A_{n}) be in C_{ρ} and assume that ω_{ρ}(A) < 1. Then there is r_{0} ∈ (0, 1) such that ω_{ρ}(\frac{1}{r_{0}} A) = \frac{1}{r_{0}} ω_{ρ}(A) < 1. Consequently, \frac{1}{r_{0}} A ∈ C_{ρ}.

According to Theorem 3.1, we have

ℜp(0) \frac{1 − r_{0}(2ρ − 1)}{1 + r_{0}} ≤ ℜp(A_{1}, …, A_{n}) ≤ ℜp(0) \frac{1 + r_{0}(2ρ − 1)}{1 − r_{0}}

for any noncommutative polynomial with matrix-valued coefficients p ∈ ℂ[X_{1}, …, X_{n}] ⊗ M_{m}, m ∈ ℕ, such that ℜp ≥ 0 on [B(H)^{n}]_{1}. Hence, we deduce that A ∼^{H} 0.

To prove that (ii) ⟹ (iii), assume that A ∼^{H} 0. Due to Theorem 2.5, we have r(A) < 1. Using now Theorem 2.2, we deduce that there exists c > 0 such that

(3.2)  P_{ρ}(rA, R) ≥ \frac{1}{c^{2}} P_{ρ}(0, R) = \frac{ρ}{c^{2}} I

for any r ∈ [0, 1). Since r(A) < 1, one can prove that lim_{r→1} P_{ρ}(rA, R) = P_{ρ}(A, R) in the operator norm topology. Consequently, taking r → 1 in relation (3.2), we obtain item (iii).

It remains to show that (iii) ⟹ (i). Assume that r(A_{1}, …, A_{n}) < 1 and P_{ρ}(A, R) ≥ aI for some constant a > 0. Note that there exists t_{0} ∈ (0, 1) such that the map

t ↦ (I − ∑_{i=1}^{n} A_{i}^{∗} ⊗ tR_{i})^{−1} + (ρ − 2)I + (I − ∑_{i=1}^{n} A_{i} ⊗ tR_{i}^{∗})^{−1}

is well-defined and continuous on [0, 1 + t_{0}] in the operator norm topology. In particular, there is ϵ_{0} ∈ (0, t_{0}) such that

∥P_{ρ}(A, R) − P_{ρ}(A, tR)∥ < \frac{a}{2}

for any t ∈ (1 − ϵ_{0}, 1 + ϵ_{0}). Consequently, if γ_{0} ∈ (1, 1 + ϵ_{0}), then

P_{ρ}(γ_{0}A, R) ≥ P_{ρ}(A, R) − ∥P_{ρ}(A, R) − P_{ρ}(γ_{0}A, R)∥I ≥ \frac{a}{2}I > 0.

Due to Theorem 1.1, we have γ_{0}A ∈ C_{ρ}, which implies ω(γ_{0}A) ≤ 1. Therefore, ω(A) ≤ \frac{1}{γ_{0}} < 1 and item (i) holds. The proof is complete. □

We remark that, when n = 1, we recover a result obtain by Foiaş [15] if ρ = 1, and by Cassier and Suciu [9] if ρ > 0.

Given A, B ∈ C_{ρ}, ρ > 0, in the same Harnack part of C_{ρ}, i.e., A ∼^{H} B, we introduce

(3.3)  Λ_{ρ}(A, B) := inf {c > 1 : A ∼_{c}^{H} B} .

Note that, due to Theorem 2.2, A ∼^{H} B if and only if the operator L_{B,A} is invertible. In this case, L_{B,A}^{−1} = L_{A,B} and

Λ_{ρ}(A, B) = max {∥L_{A,B}∥, ∥L_{B,A}∥} .

To prove the latter equality, assume that A∼_{c}^{H} B for some c ≥ 1. Due to the same theorem, we have ∥L_{B,A}∥ ≤ c and ∥L_{A,B}∥ ≤ c. Consequently,

(3.4)  max {∥L_{A,B}∥, ∥L_{B,A}∥} ≤ inf {c ≥ 1 : A∼_{c}^{H} B} = Λ_{ρ}(A, B).

On the other hand, setting c_{0} := ∥L_{B,A}∥ and c′_{0} := ∥L_{A,B}∥, Theorem

2.2 implies A≺_{c_{0}}^{H} B and B≺_{c′_{0}}^{H} A. Hence, we deduce that A∼_{d}^{H} B, where d :=

max{c_{0}, c′_{0}}. Consequently, Λ_{ρ}(A, B) ≤ d, which together with relation (3.4) imply Λ_{ρ}(A, B) = max {∥L_{A,B}∥, ∥L_{B,A}∥}, which proves our assertion. Now, we can introduce a hyperbolic (Poincaré-Bergman type) metric δ_{ρ} : Δ × Δ → ℝ^{+} on any Harnack part Δ of C_{ρ}, by setting

(3.5)  δ_{ρ}(A, B) := ln Λ_{ρ}(A, B),  A, B ∈ Δ.

Due to our discussion above, we also have

δ_{ρ}(A, B) = ln max {∥L_{A,B}∥ , ∥L_{A,B}^{−1}∥} .

Proposition 3.3. δ_{ρ} is a metric on any Harnack part of C_{ρ}.

Proof. The proof is similar to that of Proposition 2.2 from [49], but uses ρpluriharmonic kernels. □

We remark that, according to Theorem 3.2, the set

[C_{ρ}]_{<1} := {(X_{1}, …, X_{n}) ∈ B(H)^{n} : ω_{ρ}(X_{1}, …, X_{n}) < 1}

is the Harnack part of C_{ρ} containing 0. In what follows we calculate the norm of L_{Y,X} with X, Y ∈ [C_{ρ}]_{<1}, in terms of the reconstruction operators.

Theorem 3.4. If X, Y ∈ [C_{ρ}]_{<1}, then ∥L_{Y,X}∥ = ∥C_{ρ,X}C_{ρ,Y}^{−1} ∥, where

\begin{aligned}C_{ρ,X} & := Δ_{ρ,X}(I − R_{X})^{−1}, \\ Δ_{ρ,X} & := [ρI + (1 − ρ)(R_{X}^{∗} + R_{X}) + (ρ − 2)R_{X}^{∗}R_{X}]^{1/2} . \end{aligned}

Moreover, if X, Y ∈ C_{ρ} is such that X ≺^{H} Y, then ∥L_{Y,X}∥ = sup_{r∈[0,1)} ∥C_{ρ,rX}C_{ρ,rY}^{−1} ∥.

Proof. Since X, Y ∈ [C_{ρ}]_{<1}, Theorem 3.2 implies X ∼^{H} Y, r(X) < 1, and r(Y) < 1. Let c > 1 and assume that P_{ρ}(rX, R) ≤ c^{2}P_{ρ}(rY, R) for any r ∈ [0, 1). Since r(X) < 1 and r(Y) < 1, we can take the limit, as r → 1, in the operator norm topology, and obtain P_{ρ}(X, R) ≤ c^{2}P_{ρ}(Y, R). Conversely, if the latter inequality holds, then P_{ρ}(X, S) ≤ c^{2}P_{ρ}(Y, S), where S := (S_{1}, …, S_{n}) is the n-tuple of left creation operators. Applying the noncommutative Poisson transform id⊗P_{rR}, r ∈ [0, 1), and taking into account that it is a positive map, we deduce that P_{ρ}(rX, R) ≤ c^{2}P_{ρ}(rY, R) for any r ∈ [0, 1). Therefore, due to Theorem 2.2, we have

(3.6)  P_{ρ}(X, R) ≤ c^{2}P_{ρ}(Y, R) if and only if ∥L_{Y,X}∥ ≤ c.

We recall that the free pluriharmonic kernel P_{ρ}(X, R) with X ∈ [C_{ρ}]_{<1}, has the factorization P (X, R) = C_{ρ,X}^{∗}C_{ρ,X}. Due to Theorem 3.2, P_{ρ}(X, R) is invertible and, consequently, so is C_{ρ,X}. Consequently,

P_{ρ}(X, R) ≤ c^{2}P_{ρ}(Y, R) if and only if C_{ρ,Y}^{∗} ^{−1}C_{ρ,X}^{∗}C_{ρ,X}C_{ρ,Y}^{−1} ≤ c^{2}I.

Setting c_{0} := ∥C_{ρ,X}C_{ρ,Y}^{−1} ∥, we have P_{ρ}(X, R) ≤ c_{0}^{2}P_{ρ}(Y, R). Now, due to relation (3.6), we obtain

∥L_{Y,X}∥ ≤ c_{0} = ∥C_{ρ,X}C_{ρ,Y}^{−1} ∥.

Setting c′_{0} := ∥L_{Y,X}∥ and using again (3.6), we obtain P_{ρ}(X, R) ≤ c′_{0}^{2}P_{ρ}(Y, R). Hence, we deduce that C_{ρ,Y}^{∗} ^{−1}C_{ρ,X}^{∗}C_{ρ,X}C_{ρ,Y}^{−1} ≤ c′_{0}^{2}I, which implies

∥C_{ρ,X}C_{ρ,Y}^{−1} ∥ ≤ c′_{0} = ∥L_{Y,X}∥.

Therefore, ∥L_{Y,X}∥ = ∥C_{ρ,X}C_{ρ,Y}^{−1} ∥. The last part of the theorem is now obvious. □

Combining Theorem 3.4 with the remarks preceding Proposition 3.3, we obtain a concrete formula for the hyperbolic metric δ_{ρ} on [C_{ρ}]_{<1} in terms of the reconstruction operator, which is the main result of this section.

Theorem 3.5. Let δ_{ρ} : [C_{ρ}]_{<1} × [C_{ρ}]_{<1} → [0, ∞) be the hyperbolic metric. If X, Y ∈ [C_{ρ}]_{<1}, then

δ_{ρ}(X, Y) = ln max {∥C_{ρ,X}C_{ρ,Y}^{−1} ∥ , ∥C_{ρ,Y} C_{ρ,X}^{−1} ∥} ,

where

\begin{aligned}C_{ρ,X} & := Δ_{ρ,X}(I − R_{X})^{−1}, \\ Δ_{ρ,X} & := [ρI + (1 − ρ)(R_{X}^{∗} + R_{X}) + (ρ − 2)R_{X}^{∗}R_{X}]^{1/2} , \end{aligned}

and R_{X} := X_{1}^{∗} ⊗ R_{1} + ⋯ + X_{n}^{∗} ⊗ R_{n} is the reconstruction operator associated with the right creation operators R_{1}, …, R_{n} and X := (X_{1}, …, X_{n}) ∈ [C_{ρ}]_{<1}.

Using Theorem 2.2, one can easily obtain the following result. Since the proof is similar to that of Lemma 2.6 from [49], we shall omit it.

Lemma 3.6. Let X := (X_{1}, …, X_{n}) and Y := (Y_{1}, …, Y_{n}) be in C_{ρ}. Then the following properties hold.

(i) X ∼^{H} Y if and only if rX ∼^{H} rX for any r ∈ [0, 1) and sup_{r∈[0,1)} Λ_{ρ}(rX, rY) < ∞. In this case,

Λ_{ρ}(X, Y) = sup_{r∈[0,1)} Λ_{ρ}(rX, rY) and δ_{ρ}(X, Y) = sup_{r∈[0,1)} δ_{ρ}(rX, rY).

(ii) If X ∼^{H} Y, then the functions r ↦ Λ_{ρ}(rX, rY) and r ↦ δ_{ρ}(rX, rY) are increasing on [0, 1).

Putting together Theorem 3.5 and Lemma 3.6, we deduce the following result.

Theorem 3.7. Let X := (X_{1}, …, X_{n}) and Y := (Y_{1}, …, Y_{n}) be in C_{ρ} such that X ∼^{H} Y. Then the metric δ_{ρ} satisfies the relation

δ_{ρ}(X, Y) = ln max { sup_{r∈[0,1)} ∥C_{ρ,rX}C_{ρ,rY}^{−1} ∥ , sup_{r∈[0,1)} ∥C_{ρ,rY} C_{ρ,rX}^{−1} ∥} ,

where C_{ρ,X} := Δ_{ρ,X}(I − R_{X})^{−1} and R_{X} := X_{1}^{∗} ⊗ R_{1} + ⋯ + X_{n}^{∗} ⊗ R_{n} is the reconstruction operator.

Using the Harnack type inequality of Theorem 3.1, we obtain an upper bound for the hyperbolic distance δ_{ρ} on [C_{ρ}]_{<1}. First, we need the following result.

Proposition 3.8. Let f be in the noncommutative disc algebra A_{n} such that ℜf ≥ 0 and let X := (X_{1}, …, X_{n}) ∈ C_{ρ} be with ω_{ρ}(X) < 1. Then

ρ\frac{1 − ω_{ρ}(X)}{1 + ω_{ρ}(X)}ℜf(0) ≤ ℜf(X_{1}, …, X_{n}) + (ρ − 1)ℜf(0) ≤ ρ\frac{1 + ω_{ρ}(X)}{1 − ω_{ρ}(X)}.

Proof. Let r := ω_{ρ}(X) and define Y := \frac{1}{r}X. Since ω_{ρ}(Y) = \frac{1}{r}ω_{ρ}(X) = 1, we deduce that Y ∈ C_{ρ}. Applying Theorem 3.1 to Y, we obtain

\frac{1 − ω_{ρ}(X)(2ρ − 1)}{1 + ω_{ρ}(X)}ℜf(0) ≤ ℜf(X_{1}, …, X_{n}) ≤ ρ\frac{1 + ω_{ρ}(X)(2ρ − 1)}{1 − ω_{ρ}(X)}.

It is easy to see that the latter inequality is equivalent to the one from the proposition. □

Now, we can deduce the following upper bound for the hyperbolic distance on [C_{ρ}]_{<1}.

Corollary 3.9. For any X, Y ∈ [C_{ρ}]_{<1},

δ_{ρ}(X, Y) ≤ \frac{1}{2} ln \frac{(1 + ω_{ρ}(X))(1 + ω_{ρ}(Y))}{(1 − ω_{ρ}(X))(1 − ω_{ρ}(Y))}.

Documenta Mathematica 14 (2009) 595–651

Proof. Using Theorem 2.2 and the inequality of Proposition 3.8, we deduce that

Λ_{ρ}(X, 0) ≤ (\frac{1 + ω_{ρ}(X)}{1 − ω_{ρ}(X)})^{1/2} .

On the other hand, since δ_{ρ} is a metric on [C_{ρ}]_{<1}, we have δ(X, Y) ≤ δ(X, 0) + δ_{ρ}(Y, 0). Taking into account that δ_{ρ}(X, Y) = ln Λ_{ρ}(X, Y), the result follows.

□

We remark that when ρ = 1, the inequality of Corollary 3.9 is sharper then the one obtained in Corollary 2.5 from [49]. Using Corollary 2.4, on can easily obtain the following result.

Corollary 3.10. Let ρ > 0, and 1 ≤ m < n. Consider two n-tuples A := (A_{1}, …, A_{m}) ∈ B(H)^{m} and B := (B_{1}, …, B_{m}) ∈ B(H)^{m} in the class C_{ρ} and their extensions Ã:= (A_{1}, …, A_{m}, 0, …, 0) and B̃ := (B_{1}, …, B_{m}, 0, …, 0) in B(H)^{n}, respectively. Then

A∼^{H} B if and only if Ã∼^{H} B̃.

Moreover, in this case,

δ_{ρ}(A, B) = δ_{ρ}(Ã, B̃).

In what follows we provide a few properties for the map ρ ↦ δ_{ρ}(A, B).

Lemma 3.11. Let A := (A_{1}, …, A_{n}) ∈ B(H)^{n} and B := (B_{1}, …, B_{n}) ∈ B(H)^{n} be in the class C_{ρ} and let c > 0 and 0 < ρ ≤ ρ′. Then the following statements hold.

(i) if A≺_{c}^{H} B in C_{ρ}, then A≺_{c}^{H} B in C_{ρ′};

(ii) if A ∼_{c}^{H} B in C_{ρ}, then if A ∼_{c}^{H} B in C_{ρ} and

δ_{ρ′}(A, B) ≤ δ_{ρ}(A, B).

Proof. First recall that C_{ρ} ⊆ C_{ρ′}. If A≺_{c}^{H} B in C_{ρ}, then

ℜp(A_{1}, …, A_{n}) + (ρ − 1)ℜp(0) ≤ c^{2} [ℜp(B_{1}, …, B_{n}) + (ρ − 1)ℜp(0)]

for any noncommutative polynomial with matrix-valued coefficients p ∈ ℂ[X_{1}, …, X_{n}] ⊗ M_{m}, m ∈ ℕ, such that ℜp(X) ≥ 0 for any X ∈ [B(H)^{n}]_{1}. Hence, c ≥ 1 and, consequently, the inequality above holds when we replace ρ

with ρ′ ≥ ρ. This shows that A≺_{c}^{H} B in C_{ρ′}. Part (ii) is a clear consequence of

(i) and the definition of the hyperbolic metric. □

If A := (A_{1}, …, A_{n}) ∈ B(H)^{n} is a nonzero n-tuple of operators such that A ∈ [C_{∞}]_{<1}, i.e., the joint spectral radius r(A) < 1, then

ρ_{A} := inf{ρ > 0 : A ∈ C_{ρ}} > 0.

Indeed, if ρ, ρ′ ∈ (0, ∞], ρ ≤ ρ′, then C_{ρ} ⊆ C_{ρ′} and, moreover, we have

ω_{ρ′}(A) ≤ ω_{ρ}(A), r(A) = lim_{ρ→∞} ω_{ρ}(A), A ∈ B(H)^{n}.

Consequently, there exists ρ > 0 such that ω_{ρ′}(A) < 1, for any ρ′ ≥ ρ. Assume now that ρ_{A} = 0. Then T ∈ C_{ρ}, i.e., ω_{ρ}(A) ≤ 1 for any ρ > 0. On the other hand, we know that ∥A∥ ≤ ρω_{ρ}(A). Taking ρ → 0, we deduce that A = 0, which is a contradiction. This proves our assertion.

Note that if A, B ∈ [C_{∞}]_{<1}, then

ρ_{A,B} := inf{ρ > 0 : A, B ∈ C_{ρ}} = max{ρ_{A}, ρ_{B}}.

Proposition 3.12. If A, B ∈ [C_{∞}]_{<1}, then the map

[ρ_{A,B}, ∞) ∋ ρ ↦ δ_{ρ}(A, B) ∈ ℝ^{+}

is continuous, decreasing, and

lim_{ρ→∞} δ_{ρ}(A, B) = 0.

Proof. Using Theorem 3.5 and Lemma 3.11, one can easily deduce that the map ρ ↦ δ_{ρ}(A, B) is continuous and decreasing. To prove the last part of the proposition, note that since δ_{ρ}(A, B) ≤ δ_{ρ}(A, 0)+δ_{ρ}(0, B), it is enough to show that lim_{ρ→∞} δ_{ρ}(A, 0) = 0. To this end, note that Theorem 3.5, implies

(3.7)  δ_{ρ}(A, 0) = ln max {∥C_{ρ,A}C_{ρ,0}^{−1}∥ , ∥C_{ρ,0}C_{ρ,A}^{−1}∥} ,

where

C_{ρ,A}C_{ρ,0}^{−1} = \frac{1}{\sqrt{ρ}} [ρI + (1 − ρ)(R_{A}^{∗} + R_{A}) + (ρ − 2)R_{A}^{∗}R_{A}]^{1/2} (I − R_{A})^{−1}.

Hence, we deduce that

lim_{ρ→∞} ∥C_{ρ,A}C_{ρ,0}^{−1}∥ = ∥[I − (R_{A}^{∗} + R_{A}) + R_{A}^{∗}R_{A}]^{1/2} (I − R_{A})^{−1}∥

= ∥(I − R_{A}^{∗})^{−1} [I − (R_{A}^{∗} + R_{A}) + R_{A}^{∗}R_{A}] (I − R_{A})^{−1}∥

= ∥(I − R_{A}^{∗})^{−1}(I − R_{A}^{∗})(I − R_{A})(I − R_{A})^{−1}∥

= 1

Similarly, we have lim_{ρ→∞} ∥C_{ρ,0}C_{ρ,A}^{−1}∥ = 1. Using now relation (3.7), we complete the proof. □

4. Mapping theorems for free holomorphic functions on noncommutative balls

In this section, we provide mapping theorems, spectral von Neumann inequalities, and Schwarz type results for free holomorphic functions on noncommutative balls, with respect to the hyperbolic metric and the operator radius ω_{ρ}, ρ ∈ (0, ∞]. First, we prove the following mapping theorem for the classes C_{ρ}, ρ > 0.

Theorem 4.1. Let f := (f_{1}, …, f_{m}) be a contractive free holomorphic function with ∥f(0)∥ < 1 such that the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n}. If (T_{1}, …, T_{n}) ∈ B(H)^{n} is of class C_{ρ}, ρ > 0, then f(T_{1}, …, T_{n}) is of class C_{ρ_{f}}, where

(4.1)  ρ_{f} := \begin{cases}1 + (ρ − 1)\frac{1−∥f(0)∥}{1+∥f(0)∥} & if ρ < 1 \\ 1 + (ρ − 1)\frac{1+∥f(0)∥}{1−∥f(0)∥} & if ρ ≥ 1. \end{cases}

Proof. Let p ∈ ℂ[Z_{1}, …, Z_{m}] ⊗ M_{k}, k ∈ ℕ, be such that ℜp ≥ 0 on the unit ball [B(H)^{m}]_{1}. This is equivalent to ℜp(S′_{1}, … S′_{m}) ≥ 0, where S′_{1}, …, S′_{m} are the left creation operators on the full Fock space F^{2}(H_{m}). Applying the noncommutative Poisson transform P_{f(X_{1},...,X_{n})} ⊗ id, which is a completely positive linear map, to the inequality ℜp(S′_{1}, … S′_{m}) ≥ 0, we obtain

ℜp(f(X_{1}, …, X_{n})) ≥ 0, X ∈ [B(H)^{n}]_{1}.

Moreover, since the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n}, we deduce that the boundary function of the composition p∘f is p(f̃_{1}, …, f̃_{m}) ∈ A_{n} ¯⊗_{min}M_{k}. Assume that (T_{1}, …, T_{n}) ∈ C_{ρ}. Using the free pluriharmonic functional calculus of Theorem 1.3 and Theorem 1.1, we deduce that

(4.2)  ℜ(p ∘ f)(T_{1}, …, T_{n}) + (ρ − 1)ℜ(p ∘ f)(0) ≥ 0.

On the other hand, according to the Harnack type inequality of Theorem 1.4 from [49] applied to the positive free pluriharmonic function ℜp at the point f(0) = (f_{1}(0), …, f_{m}(0)), we have

(4.3)  ℜp(0)\frac{1 − ∥f(0)∥}{1 + ∥f(0)∥} ≤ ℜp(f(0)) ≤ ℜp(0)\frac{1 + ∥f(0)∥}{1 − ∥f(0)∥}.

Let γ > 0 and note that

(4.4)  ℜp(f(T_{1}, …, T_{n})) + (γ − 1)ℜp(0) = A + B,

where

\begin{aligned}A & := ℜp(f(T_{1}, …, T_{n})) + (ρ − 1)p(f(0)) \\ B & := (γ − 1)ℜp(0) − (ρ − 1)p(f(0)). \end{aligned} ; (4.5)

Assume now that ρ ≥ 1. Using the second inequality in (4.3), we obtain

\begin{aligned}B & ≥ (γ − 1)ℜp(0) − (ρ − 1)ℜp(0)\frac{1 + ∥f(0)∥}{1 − ∥f(0)∥} \\ & = [(γ − 1) − (ρ − 1)\frac{1 + ∥f(0)∥}{1 − ∥f(0)∥}] ℜp(0), \end{aligned}

which is positive if γ ≥ 1 + (ρ − 1)\frac{1+∥f(0)∥}{1−∥f(0)∥}. In this case, using relation (4.4) and (4.2), we obtain

ℜp(f(T_{1}, …, T_{n})) + (γ − 1)ℜp(0) ≥ 0

for any p ∈ ℂ[Z_{1}, …, Z_{n}] ⊗ M_{k}, k ∈ ℕ, be such that ℜp ≥ 0 on the unit ball [B(H)^{m}]_{1}. Applying Theorem 1.1, we deduce that f(T_{1}, …, T_{n}) ∈ C_{γ}. In particular, we have f(T_{1}, …, T_{n}) ∈ C_{δ_{f}} where

δ_{f} := 1 + (ρ − 1)\frac{1 + ∥f(0)∥}{1 − ∥f(0)∥}.

Now, we consider the case ρ ∈ (0, 1). Using the first inequality in (4.3), we obtain

B ≥ [(γ − 1) − (ρ − 1)\frac{1 − ∥f(0)∥}{1 + ∥f(0)∥}] ℜp(0),

which is positive if γ ≥ 1 + (ρ − 1)\frac{1−∥f(0)∥}{1+∥f(0)∥}. As above, using relations (4.4) and (4.2), we obtain

ℜp(f(T_{1}, …, T_{n})) + (γ − 1)ℜp(0) ≥ 0

for any p ∈ ℂ[Z_{1}, …, Z_{n}] ⊗ M_{k}, k ∈ ℕ, be such that ℜp ≥ 0 on the unit ball [B(H)^{m}]_{1}. Theorem 1.1 implies f(T_{1}, …, T_{n}) ∈ C_{γ}. In particular, we have f(T_{1}, …, T_{n}) ∈ C_{δ_{f}} where

δ_{f} := 1 + (ρ − 1)\frac{1 − ∥f(0)∥}{1 + ∥f(0)∥}.

The proof is complete. □

Note that under the conditions of Theorem 4.1, ρ ≤ ρ_{f} and ρ = 1 ⟹ ρ_{f} = 1. Moreover, if ρ ≠ 1, then ρ_{f} = ρ if and only if f(0) = 0. On can also show that ρ_{f} ≤ 1 if ρ ≤ 1. We remark that, under the conditions of Theorem 4.1, there exists T := (T_{1}, …, T_{n}) ∈ B(H)^{n} such that if ρ > 0 is the smallest positive number such that (T_{1}, …, T_{n}) ∈ C_{ρ}, then there exists a free holomorphic function f such that ρ_{f} is the smallest positive number with the property that f(T_{1}, …, T_{n}) ∈ C_{ρ_{f}}. Indeed, if n ≤ m, take f(X_{1}, …, X_{n}) = (X_{1}, …, X_{n}, 0, …, 0) and use Corollary 2.4. When n > m, take f(X_{1}, …, X_{n}) = (X_{1}, …, X_{m}) and T := (T_{1}, …, T_{n}, 0, …, 0) with (T_{1}, …, T_{n}) ∈ C_{ρ}.

Corollary 4.2. Let f := (f_{1}, …, f_{m}) be a bounded free holomorphic function with ∥f(0)∥ < ∥f∥_{∞} such that the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n}. If (T_{1}, …, T_{n}) ∈ B(H)^{n} is of class C_{ρ}, ρ > 0, then

ω_{ρ_{f}} (f(T_{1}, …, T_{n})) ≤ ∥f∥_{∞},

where ρ_{f} is given by relation (4.1). In particular, if f(0) = 0 and (T_{1}, …, T_{n}) ∈ C_{ρ}, then

ω_{ρ}(f(T_{1}, …, T_{n})) ≤ ∥f∥_{∞}.

Proof. Applying Theorem 4.1 the function \frac{1}{∥f∥_{∞}} f, we deduce that \frac{1}{∥f∥_{∞}} f(T_{1}, …, T_{n}) is in the class C_{ρ_{f}}, which is equivalent to

ω_{ρ_{f}} (\frac{1}{∥f∥_{∞}} f(T_{1}, …, T_{n})) ≤ 1, and the first inequality of the theorem follows. Hence, and using the fact that ρ_{f} = ρ when f(0) = 0, we complete the proof. □

A simple consequence of Corollary 4.2 is the following power inequality.

Corollary 4.3. If (T_{1}, …, T_{n}) ∈ B(H)^{n} is non-zero, ρ ∈ (0, ∞), and k ≥ 1, then

ω_{ρ}(T_{α} : α ∈ 𝔽_{n}^{+}, |α| = k) ≤ ω_{ρ}(T_{1}, …, T_{n}).

Proof. Since ∥(T_{1}, …, T_{n})∥ ≤ ρω_{ρ}(T_{1}, …, T_{n}), we have ω_{ρ}(T_{1}, …, T_{n}) ≠ 0. Applying the second part of Corollary 4.2 to the n-tuple of operators (\frac{1}{ω_{ρ}(T_{1},...,T_{n})}T_{1}, …, \frac{1}{ω_{ρ}(T_{1},...,T_{n})}T_{n}) ∈ C_{ρ} and to the free holomorphic function

f(X_{1}, …, X_{n}) := (X_{α} : α ∈ 𝔽_{n}^{+}, |α| = k), (X_{1}, …, X_{n}) ∈ [B(H)^{n}]_{1},

we complete the proof. □

Theorem 4.4. Let f := (f_{1}, …, f_{m}) be a bounded free holomorphic function with ∥f(0)∥ < ∥f∥_{∞} such that the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n}. Then, for each r ∈ [0, 1),

sup_{T ∈C_{ρ}, ω_{ρ}(T)≤r} ω_{ρ_{f}} (f(T_{1}, …, T_{n})) ≤ ∥f(rS_{1}, …, rS_{n})∥,

where S_{1}, …, S_{n} are the left creation operators.

Proof. Consider the free holomorphic function f_{r}, defined by

f_{r}(X_{1}, …, X_{n}) := f(rX_{1}, …, rX_{n}), (X_{1}, …, X_{n}) ∈ [B(H)^{n}]_{1}

and recall that ∥f_{r}∥_{∞} = ∥f(rS_{1}, …, rS_{n})∥. Applying Corollary 4.2 to f_{r}, we have

(4.6)  ω_{ρ_{fr}} (f_{r}(A_{1}, …, A_{n})) ≤ ∥f_{r}∥_{∞}, (A_{1}, …, A_{n}) ∈ C_{ρ}

Since f(0) = f_{r}(0), we have ρ_{f} = ρ_{f_{r}}. Consequently, if we assume that ω_{ρ}(T_{1}, …, T_{n}) ≤ r < 1, then (\frac{1}{r}T_{1}, …, \frac{1}{r}T_{n}) ∈ C_{ρ} and inequality (4.6) implies

ω_{ρ_{f}} (f(T_{1}, …, T_{n})) = ω_{ρ_{f}} (f_{r} (\frac{1}{r}T_{1}, …, \frac{1}{r}T_{n})) ≤ ∥f_{r}∥_{∞},

which completes the proof. □

Corollary 4.5. Let (T_{1}, …, T_{n}) ∈ B(H)^{n} be such that ω_{ρ}(T_{1}, …, T_{n}) < 1, and let f := (f_{1}, …, f_{m}) be a bounded free holomorphic function with the following properties:

(i) the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n}.

(ii) f_{j} has the standard representation of the form

f_{j}(X_{1}, …, X_{n}) = ∑_{|α|≥k} a_{α}^{(j)}X_{α}, j = 1, …, m.

Then

ω_{ρ}(f(T_{1}, …, T_{n})) ≤ ω_{ρ}(T_{1}, …, T_{n})^{k}∥f∥_{∞}.

Proof. Consider the free holomorphic function g := \frac{1}{∥f∥_{∞}} f. Note that ∥g∥_{∞} = 1 and g(0) = 0. According to the Schwarz lemma for free holomorphic functions (see Theorem 2.4 from [44]), we have

∥g(X_{1}, …, X_{n})∥ ≤ ∥(X_{1}, …, X_{n})∥^{k}, (X_{1}, …, X_{n}) ∈ [B(H)^{n}]_{1}.

Denote r := ω_{ρ}(T_{1}, …, T_{n}) < 1, ρ > 0, and consider

g_{r}(X_{1}, …, X_{n}) := g(rX_{1}, …, rX_{n}), (X_{1}, …, X_{n}) ∈ [B(H)^{n}]_{1}.

Note that the inequality above implies ∥g_{r}∥_{∞} ≤ r^{k}. Applying now Theorem 4.4 to g, and using the latter inequality, we obtain

ω_{ρ}(g(T_{1}, …, T_{n})) ≤ ∥g_{r}∥_{∞} ≤ r^{k} = ω_{ρ}(T_{1}, …, T_{n})^{k}.

Hence, the result follows. □

Corollary 4.6. Let (T_{1}, …, T_{n}) ∈ B(H)^{n} be such that ω_{ρ}(T_{1}, …, T_{n}) < 1, and let f : [B(H)^{n}]_{1} → B(H) be a free holomorphic function with ℜf ≤ I and having the standard representation

f(X_{1}, …, X_{n}) = ∑_{|α|≥k} a_{α}X_{α}, where k ≥ 1.

Then

ω_{ρ}(f(T_{1}, …, T_{n})) ≤ \frac{2ω_{ρ}(T_{1}, …, T_{n})^{k}}{1 − ω_{ρ}(T_{1}, …, T_{n})^{k}} .

Documenta Mathematica 14 (2009) 595–651

Proof. According to the Carathéodory type result for free holomorhic functions (see Theorem 5.1 from [51]), we have

∥f(X_{1}, …, X_{n})∥ ≤ \frac{2∥ ∑_{|β|=k} X_{β}X_{β}^{∗}∥^{1/2}}{1 − ∥ ∑_{|β|=k} X_{β}X_{β}^{∗}∥^{1/2}} , (X_{1}, …, X_{n}) ∈ [B(H)^{n}]_{1}.

Hence, we deduce that ∥f_{r}∥_{∞} ≤ \frac{2r^{k}}{1−r^{k}} for any r ∈ (0, 1). Setting r := ω_{ρ}(T_{1}, …, T_{n}) < 1, ρ > 0, and applying Theorem 4.4, we obtain

ω_{ρ}(f(T_{1}, …, T_{n})) ≤ ∥f_{r}∥_{∞} ≤ \frac{2ω_{ρ}(T_{1}, …, T_{n})^{k}}{1 − ω_{ρ}(T_{1}, …, T_{n})^{k}} .

The proof is complete. □

Lemma 4.7. Let f := (f_{1}, …, f_{m}) be a contractive free holomorphic function with ∥f(0)∥ < 1 such that the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n}. Let A := (A_{1}, …, A_{n}) ∈ B(H)^{n} and B := (B_{1}, …, B_{n}) ∈ B(H)^{n} be in the class C_{ρ} ⊂ B(H)^{n} and let c ≥ 1. If

A≺_{c}^{H} B, then f(A) and f(B) are in C_{ρ_{f}} ⊂ B(H)^{m} and f(A)≺_{c}^{H} f(B), where ρ_{f} is

given by relation (4.1).

Proof. First, note that, due to Theorem 4.1, f(A), f(B) are in C_{ρ_{f}}, where ρ_{f} is given by relation (4.1). Let p ∈ ℂ[Z_{1}, …, Z_{m}] ⊗ M_{k}, k ∈ ℕ, be such that ℜp ≥ 0 on the unit ball [B(H)^{m}]_{1}. According to the proof of Theorem 4.1, the boundary function of the composition p ∘ f is p(f̃_{1}, …, f̃_{m}) ∈ A_{n} ¯⊗_{min}M_{k} and ℜ(p ∘ f) ≥ 0. Using the free pluriharmonic functional calculus for the class C_{ρ}

and Theorem 2.2, if A, B are in C_{ρ} and A≺_{c}^{H} B, c ≥ 1, then

\begin{matrix}ℜ(p ∘ f) & (A_{1}, …, A_{n}) + (ρ − 1)ℜ(p ∘ f)(0) \\ ≤ c^{2} [ℜ(p ∘ f)(B_{1}, …, B_{n}) + (ρ − 1)ℜ(p ∘ f)(0)] . \end{matrix} ; (4.7)

Assume now that ρ ≥ 1. Due to the Harnack type inequality (4.3), the inequality (4.7) implies

ℜ(p∘f)(A_{1}, …, A_{n}) ≤ c^{2}ℜ(p∘f)(B_{1}, …, B_{n})+(c^{2} −1)(ρ−1)ℜp(0)\frac{1 + ∥f(0)∥}{1 − ∥f(0)∥},

which is equivalent to

\begin{matrix}ℜ(p ∘ f) & (A_{1}, …, A_{n}) + (ρ_{f} − 1)ℜ(p ∘ f)(0) \\ ≤ c^{2} [ℜ(p ∘ f)(B_{1}, …, B_{n}) + (ρ_{f} − 1)ℜ(p ∘ f)(0)] , \end{matrix}

where δ_{f} := 1 + (ρ − 1)\frac{1+∥f(0)∥}{1−∥f(0)∥}. Applying Theorem 2.2, we deduce that

f(A)≺_{c}^{H} f(B).

Now, we consider the case ρ ∈ (0, 1). The inequality (4.7) and the Harnack type inequality (4.3) imply

ℜ(p∘f)(A_{1}, …, A_{n}) ≤ c^{2}ℜ(p∘f)(B_{1}, …, B_{n})+(c^{2} −1)(ρ−1)ℜp(0)\frac{1 − ∥f(0)∥}{1 + ∥f(0)∥}.

As above, we deduce that f(A)≺_{c}^{H} f(B) in C_{ρ_{f}}, where δ_{f} := 1 + (ρ − 1)\frac{1−∥f(0)∥}{1+∥f(0)∥}. This completes the proof. □

Theorem 4.8. Let δ_{ρ} : Δ × Δ → [0, ∞) be the hyperbolic metric on a Harnack part Δ of C_{ρ}, and let f := (f_{1}, …, f_{m}) be a contractive free holomorphic function with ∥f(0)∥ < 1 such that the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n}.Then

δ_{ρ_{f}} (f(A), f(B)) ≤ δ_{ρ}(A, B), A, B ∈ Δ,

where ρ_{f} is given by relation (4.1).

Proof. Let A, B ∈ Δ ⊂ C_{ρ}, i.e., there is c ≥ 1 such that A∼_{c}^{H} B. According to

Theorem 4.1 and Lemma 4.7, f(A) and f(B) are in C_{ρ_{f}}, and f(A)∼_{c}^{H} f(B) in C_{ρ_{f}}, where ρ_{f} is given by relation (4.1). Hence and taking into account that

δ_{ρ}(A, B) := ln inf {c > 1 : A ∼_{c}^{H} B} , A, B ∈ Δ,

we deduce that

δ_{ρ_{f}} (f(A), f(B)) ≤ δ_{ρ}(A, B), A, B ∈ Δ.

The proof is complete. □

Now, we can deduce the following Schwarz type result.

Corollary 4.9. Let δ_{ρ} : Δ × Δ → [0, ∞) be the hyperbolic metric on a Harnack part Δ of C_{ρ}, and let f := (f_{1}, …, f_{m}) be a contractive free holomorphic function with f(0) = 0 such that the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n}. Then

δ_{ρ}(f(A), f(B)) ≤ δ_{ρ}(A, B), A, B ∈ Δ.

We recall that, due to Theorem 3.2, the open ball [C_{ρ}]_{<1} is the Harnack part of C_{ρ} containing 0. Consequently, Theorem 4.8 and Corollary 4.9 hold in the particular case when Δ := [C_{ρ}]_{<1}. Ky Fan [14] showed that the von Neumann inequality [57] is equivalent to the fact that if T ∈ B(H) is a strict contraction (∥T ∥ < 1) and f : 𝔻 → 𝔻 is an analytic function, then ∥f(T)∥ < 1. A multivariable analogue of this result was obtained in [51]. In what follows, we provide a spectral version of this result, when the norm is replaced by the joint spectral radius.

Theorem 4.10. Let f := (f_{1}, …, f_{m}) be a contractive free holomorphic function with ∥f(0)∥ < 1 such that the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n}. If (T_{1}, …, T_{n}) ∈ B(H)^{n} and the joint spectral radius r(T_{1}, …, T_{n}) < 1, then

r(f(T_{1}, …, T_{n})) < 1.

Proof. Assume that (T_{1}, …, T_{n}) ∈ B(H)^{n} has the joint spectral radius r(T_{1}, …, T_{n}) < 1. Taking into account that r(T_{1}, …, T_{n}) = lim_{ρ→∞} ω_{ρ}(T_{1}, …, T_{n}), we find δ > 1 such that ω_{ρ}(T_{1}, …, T_{n}) < 1. Therefore, we have T := (T_{1}, …, T_{n}) ∈ C_{ρ} and, due to Theorem 3.2, the n-tuple T

is Harnack equivalent to 0. Consequently, T ≺_{c}^{H} 0 for some constant c ≥ 1.

According to Theorem 4.1, f(T) and f(0) are in the class C_{ρ_{f}}, where ρ_{f} is

given by relation (4.1). On the other hand, Lemma 4.7 implies f(T)≺_{c}^{H} f(0) in

C_{ρ_{f}}. Since ∥f(0)∥ < 1, we have the joint spectral radius r(f(0)) < 1. Applying

Theorem 2.5, we deduce that f(0)≺_{c}^{H} 0 in C_{ρ_{f}}. Therefore, we have f(T)≺_{c}^{H} 0

in C_{ρ_{f}}. Applying again Theorem 2.5, we have r(f(T)) < 1. The proof is complete. □

An analogue of Theorem 4.10 for n-tuples of operators with joint operator radius ω_{ρ}(T_{1}, …, T_{n}) < 1 is the following.

Theorem 4.11. Let f := (f_{1}, …, f_{m}) be a contractive free holomorphic function with ∥f(0)∥ < 1 such that the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n}. If (T_{1}, …, T_{n}) ∈ B(H)^{n} and ω_{ρ}(T_{1}, …, T_{n}) < 1, then

ω_{ρ_{f}} (f(T_{1}, …, T_{n})) < 1,

where ρ_{f} is defined by relation (4.1). In particular, if f(0) = 0, then ω_{ρ}(f(T_{1}, …, T_{n})) < 1.

Proof. If T := (T_{1}, …, T_{n}) ∈ B(H)^{n} and ω_{ρ}(T_{1}, …, T_{n}) < 1, then T ∈ C_{ρ}. According to Theorem 3.2, we have

r(T_{1}, …, T_{n}) < 1 and P_{ρ}(T, R) ≥ aI

for some constant a > 0. Applying Theorem 4.1 and Theorem 4.10, we deduce that f(T) ∈ C_{ρ_{f}} and r(f(T)) < 1. Since ω_{ρ}(T) < 1, Theorem 3.2 implies

T ∼^{H} 0. In particular, we have 0≺_{c}^{H} T for some constant c ≥ 1. Applying Lemma

4.7, we deduce that f(0)≺_{c}^{H} f(T) in C_{ρ_{f}}, where ρ_{f} is given by relation (4.1).

Hence, and using Theorem 2.2 (part (ii)), we get

P_{ρ_{f}} (rf(0), R) ≤ c^{2}P_{ρ_{f}} (rf(T), R), r ∈ [0, 1).

Since r(f(0)) < 1 and r(f(T)) < 1, the latter inequality implies

(4.8)  P_{ρ_{f}} (f(0), R) ≤ c^{2}P_{ρ_{f}} (f(T), R),  r ∈ [0, 1).

On the other hand, since the mapping X ↦ P_{1}(X, R) is a positive free pluriharmonic function on [B(H)^{n}]_{1}, the Harnack inequality (3.1) implies

P_{1}(f(0), R) ≥ P_{1}(0, R)\frac{1 − ∥f(0)∥}{1 + ∥f(0)∥} = \frac{1 − ∥f(0)∥}{1 + ∥f(0)∥}I.

Therefore, we have

\begin{aligned}P_{ρ_{f}} (f(0), R) & = P_{1}(f(0), R) + (ρ_{f} − 1)I \\ & ≥ (ρ_{f} − 1 + \frac{1 − ∥f(0)∥}{1 + ∥f(0)∥}) I. \end{aligned}

Since

a := ρ_{f} − 1 + \frac{1 − ∥f(0)∥}{1 + ∥f(0)∥} = \begin{cases}ρ\frac{1−∥f(0)∥}{1+∥f(0)∥} & if ρ < 1 \\ (ρ − 1)\frac{1+∥f(0)∥}{1−∥f(0)∥} + \frac{1−∥f(0)∥}{1+∥f(0)∥} & if ρ ≥ 1, \end{cases}

we have a > 0. Combining the latter inequality with (4.8) we obtain

P_{ρ_{f}} (f(T), R) ≥ \frac{a}{c^{2}} I.

Using again Theorem 3.2, we deduce that ω_{ρ_{f}} (f(T)) < 1. The last part of the theorem follows from Theorem 4.1. This completes the proof. □

Remark 4.12. If m = 1, all the results of this section remain true when the condition ∥f(0)∥ < 1 is dropped if f is a nonconstant contractive free holomorphic function with boundary function in the noncommutative algebra A_{n}.

5. Carathéodory metric on the open noncommutative ball [C_{∞}]_{<1} and Lipschitz mappings

In this section, we introduce a Carathéodory type metric d_{K} on the open ball of all n-tuples of operators (T_{1}, …, T_{n}) with joint spectral radius r(T_{1}, …, T_{n}) < 1. We obtain a concrete formula for d_{K} in terms of the free pluriharmonic kernel on the open unit ball [C_{∞}]_{<1}. This is used to prove that the metric d_{K} is complete on [C_{∞}]_{<1} and its topology coincides with the operator norm topology. We need some notation. Consider the noncommutative balls

[C_{ρ}]_{<1} := {(X_{1}, …, X_{n}) ∈ B(H)^{n} : ω_{ρ}(X_{1}, …, X_{n}) < 1} for ρ ∈ (0, ∞],

where ω_{∞}(X_{1}, …, X_{n}) := r(X_{1}, …, X_{n}) is the joint spectral radius of (X_{1}, …, X_{n}), and set

[C_{ρ}]^{≺ 0} := C_{ρ} ∩ [C_{∞}]_{<1} for ρ ∈ (0, ∞).

According to Theorem 1.35 from [48], if ρ, ρ′ ∈ (0, ∞], ρ ≤ ρ′, then C_{ρ} ⊆ C_{ρ′} and, moreover, we have

ω_{ρ′}(X) ≤ ω_{ρ}(X), r(X) = lim_{ρ→∞} ω_{ρ}(X), X ∈ B(H)^{n}.

Consequently, we have

[C_{ρ}]^{≺ 0} ⊆ [C_{ρ′}]^{≺ 0}, [C_{ρ}]_{<1} ⊆ [C_{ρ′}]_{<1}.

Due to Theorem 2.5 and Theorem 3.2, one can easily see that

{X ∈ C_{ρ} : X ∼^{H} 0} = [C_{ρ}]_{<1} ⊂ [C_{ρ}]^{≺ 0} = {X ∈ C_{ρ} : X≺^{H} 0}

for any ρ ∈ (0, ∞). Note also that

⋃_{ρ>0}[C_{ρ}]_{<1} = ⋃_{ρ>0}[C_{ρ}]^{≺ 0} = [C_{∞}]_{<1}.

Indeed, if X ∈ [C_{∞}]_{<1}, i.e., r(X) < 1, then taking into account that r(X) = lim_{ρ→∞} ω_{ρ}(X), we find ρ > 0 such that ω_{ρ}(X) < 1. Thus X ∈ [C_{ρ}]_{<1}, which proves our assertion. Note also that ⋃_{ρ>0}[C_{ρ}]_{<1} is dense (in the norm topology)

in the set C_{∞} of all n-tuples of operators (T_{1}, …, T_{n}) with joint spectral radius r(T_{1}, …, T_{n}) ≤ 1. Now, we introduce the map d_{K} : [C_{∞}]_{<1} × [C_{∞}]_{<1} → [0, ∞) by setting

(5.1)  d_{K}(A, B) = sup_{p} ∥ℜp(A) − ℜp(B)∥,  A, B ∈ [C_{∞}]_{<1},

where the supremum is taken over all polynomials p ∈ ℂ[X_{1}, …, X_{n}] ⊗ M_{m}, m ∈ ℕ, with ℜp(0) = I and ℜp ≥ 0 on [B(H)^{n}]_{1}. In what follows we will prove that d_{K} is a metric and obtain a concrete formula in terms of the free pluriharmonic kernel on the open unit ball [C_{∞}]_{<1}. First, we need the following result.

Lemma 5.1. Let G be a free pluriharmonic function on [B(H)^{n}]_{1} with coefficients in B(E), such that G(0) = I and G ≥ 0. If A, B ∈ [C_{∞}]_{<1}, then

∥G(A) − G(B)∥ ≤ ∥P_{1}(A, R) − P_{1}(B, R)∥,

where where P_{1}(X, R) is the free pluriharmonic Poisson kernel defined by

P_{1}(X, R) := ∑_{k=1}^{∞} ∑_{|α|=k} X_{α} ⊗ R∗ ; α̃ + I ⊗ I + ∑_{k=1}^{∞} ∑_{|α|=k} X_{α}^{∗} ⊗ Rα̃, X ∈ [C_{∞}]_{<1},

and the convergence is in the operator norm topology.

Proof. Since G is a positive free pluriharmonic function of [B(H)^{n}]_{1} it has a unique representation of the form

G(X_{1}, …, X_{n}) = ∑_{k=1}^{∞} ∑_{|α|=k} X_{α}^{∗}⊗A_{(α)}^{∗}+I⊗I+∑_{k=1}^{∞} ∑_{|α|=k} X_{α}⊗A_{(α)}, X ∈ [B(H)^{n}]_{1},

for some A_{(α)} ∈ B(E), where the series converge in the operator norm topology. Applying Theorem 5.2 from [47] to G, we find a completely positive linear map

μ : R_{n}^{∗} + R_{n} → B(E) with μ(I) = I and μ(R∗ ; α̃) = A_{(α)} if |α| ≥ 1.

Since A, B ∈ [C_{ρ}]_{<1}, we have r(A) < 1 and r(B) < 1. According to the free pluriharmonic functional calculus, P_{ρ}(A, R), P_{ρ}(B, R), G(A), and G(B) are well-defined and the corresponding series converge in the operator norm topology. Consequently, we have

G(A) = (id ⊗ μ)(P_{1}(A, R)) and G(A) = (id ⊗ μ)(P_{1}(A, R)).

Taking into account that μ is completely positive linear map with μ(I) = I, we have

∥G(A) − G(B)∥ ≤ ∥μ∥∥P_{1}(A, R) − P_{1}(B, R)∥ = ∥P_{1}(A, R) − P_{1}(B, R)∥.

The proof is complete. □

According to Lemma 5.1, it makes sense to define the map d′_{K} : [C_{∞}]_{<1} × [C_{∞}]_{<1} → [0, ∞) by setting

d′_{K}(A, B) := sup_{u} ∥u(A) − u(B)∥ < ∞,

where the supremum is taken over all free pluriharmonic functions u on [B(H)^{n}]_{1} with coefficients in B(E), such that u(0) = I and u ≥ 0.

Using the the free pluriharmonic functional calculus for for n-tuples of operators (T_{1}, …, T_{n}) with the joint spectral radius r(T_{1}, …, T_{n}) < 1, one can extend Proposition 3.1 from [49] and show that for any A, B ∈ [C_{∞}]_{<1},

d′_{K}(A, B) = d_{K}(A, B),

where d_{K} is defined by relation (5.1). Since the proof is essentially the same, we shall omit it.

Proposition 5.2. d_{K} is a metric on [C_{∞}]_{<1} satisfying relation

d_{K}(A, B) = ∥P_{1}(A, R) − P_{1}(B, R)∥, A, B ∈ [C_{∞}]_{<1}.

In addition, the map [0, 1) ∋ r ↦ d_{K}(rA, rB) ∈ ℝ^{+} is increasing and

d_{K}(A, B) = sup_{r∈[0,1)} d_{K}(rA, rB).

Proof. Using Lemma 5.1 we deduce that d_{K}(A, B) ≤ ∥P_{1}(A, R) − P_{1}(B, R)∥. The rest of the proof is similar to that of Proposition 3.2 from [49], so we shall omit it. □

Now, we can prove the main result of this section.

Theorem 5.3. Let d_{K} be the Carathéodory metric on [C_{∞}]_{<1}. Then the following statements hold:

(i) the d_{K}-topology coincides with the norm topology on [C_{∞}]_{<1}; (ii) [C_{ρ}]^{≺ 0} is a d_{K}-closed subset of [C_{∞}]_{<1} for any ρ > 0; (iii) the metric d_{K} is complete on [C_{∞}]_{<1}.

Proof. We recall that the free pluriharmonic Poisson kernel is given by

P_{1}(X, R) = ∑_{k=1}^{∞} ∑_{|α|=k} X_{α} ⊗ R∗ ; α̃ + I ⊗ I + ∑_{k=1}^{∞} ∑_{|α|=k} X_{α}^{∗} ⊗ Rα̃, X ∈ [C_{∞}]_{<1},

where the convergence is in the operator norm topology. Let R_{A} := A_{1}^{∗} ⊗ R_{1} + ⋯ + A_{n}^{∗} ⊗ R_{n} be the reconstruction operator. Note that, due to the noncommutative von Neumann inequality, we have

∥A − B∥ = ∥R_{A} − R_{B}∥

= ∥\frac{1}{2π} ∫_{0}^{2π} e^{it}[P_{1}(A, e^{it}R) − P_{1}(B, e^{it}R)]dt∥

≤ sup_{t∈[0,2π]} ∥P_{1}(A, e^{it}R) − P_{1}(B, e^{it}R)∥

≤ ∥P_{1}(A, R) − P_{1}(B, R)∥.

Now, Proposition 5.2 implies

(5.2)  ∥A − B∥ ≤ d_{K}(A, B),  A, B ∈ [C_{∞}]_{<1},

which shows that the d_{K}-topology is stronger then the norm topology on [C_{∞}]_{<1}. Conversely, to prove that the norm topology on [C_{∞}]_{<1} is stronger than the d_{K}-topology, note that since r(R_{A}) = r(A) < 1 and r(R_{B}) = r(B) < 1, the operators I − R_{A} and I − R_{B} are invertible. Thus

d_{K}(A, B) = ∥P_{1}(A, R) − P_{1}(B, R)∥ ≤ 2∥(I − R_{A})^{−1} − (I − R_{B})^{−1}∥

for any A, B ∈ [C_{∞}]_{<1}. Hence and due to the continuity of the maps X ↦ I − R_{X} on B(H)^{n} and Y ↦ Y^{−1} on the group of invertible elements in B(H ⊗ F^{2}(H_{n})), in the operator norm topology, we deduce our assertion. In conclusion, the d_{K}-topology coincides with the norm topology on [C_{∞}]_{<1}.

Now, to prove (ii), let {A^{(k)} := (A_{1}^{(k)}, …, A_{n}^{(k)})}_{k=1}^{∞} be a d_{K}-Cauchy sequence in [C_{ρ}]^{≺ 0} ⊂ C_{ρ}. Due to inequality (5.2), we deduce that {A^{(k)}}_{k=1}^{∞} is a Cauchy sequence in the norm topology of B(H)^{n}. Since C_{ρ} is closed in the operator norm topology, there exists T := (T_{1}, …, T_{n}) in C_{ρ} such that ∥T − A^{(k)}∥ → 0, as k → ∞. Now let us prove that the joint spectral radius r(T) < 1. Since {A^{(k)}}_{k=1}^{∞} is a d_{K}-Cauchy sequence, there exists k_{0} ∈ ℕ such that d_{K}(A^{(k)}, A^{(k_{0})}) ≤ 1 for

any k ≥ k_{0}. On the other hand, since A^{(k_{0})} ∈ [C_{ρ}]^{≺ 0}, i.e., A^{(k_{0})}≺^{H} 0, Theorem 2.2 shows that there is c ≥ 1 such that P_{ρ}(rA^{(k_{0})}, R) ≤ c^{2}δ for any r ∈ [0, 1). Hence, and due to the noncommutative von Neumann inequality, we deduce that

\begin{aligned}P_{ρ}(rA^{(k)}, R) & ≤ (∥P_{ρ}(rA^{(k)}, R) − P_{ρ}(rA^{(k_{0})}, R)∥ + ∥P_{ρ}(rA^{(k_{0})}, R)∥) I \\ & ≤ (d_{K}(A^{(k)}, A^{(k_{0})}) + ∥P_{ρ}(rA^{(k_{0})}, R)∥) I ≤ (1 + c^{2}δ)I \end{aligned} ; (5.3)

for any k ≥ k_{0} and r ∈ [0, 1).

We show now that lim_{k→∞} P_{ρ}(rA^{(k)}, R) = P_{ρ}(rT, R) in the operator norm topology. First, one can easily see that, since T, A^{(k)} ∈ C_{ρ}, we have

∑_{|α|=p} T_{α}T_{α}^{∗} ≤ ρ^{2}I and ∑_{|α|=p} A_{α}^{(k)}A_{α}^{(k)} ≤ ρ^{2}I

for any p, k = 1, 2, . . .. Given ϵ > 0 and r ∈ (0, 1), let m ∈ ℕ be such that ∑_{p=m}^{∞} ρr^{p} < \frac{ϵ}{2}. Note that

∥P (rA^{(k)}, R) − P (rT, R)∥

≤ 2 ∑_{p=1}^{m−1} ∥ ∑_{|α|=p} r^{|α|}(A_{α}^{(k)} − T_{α}) ⊗ R∗ ; α̃∥

+ 2 ∑_{p=m}^{∞} ∥ ∑_{|α|=p} r^{|α|}A_{α}^{(k)} ⊗ R∗ ; α̃∥ + 2 ∑_{p=m}^{∞} ∥ ∑_{|α|=p} r^{|α|}T_{α} ⊗ R∗ ; α̃∥

= 2 ∥ ∑_{p=1}^{m−1} r^{p} ∑_{|α|=p}(A_{α}^{(k)} − T_{α})(A_{α}^{(k)} − T_{α})^{∗}∥

+ 2 ∥ ∑_{p=1}^{m−1} r^{p} ∑_{|α|=p} A_{α}^{(k)}A_{α}^{(k)}^{∗}∥ + 2 ∥ ∑_{p=1}^{m−1} r^{p} ∑_{|α|=p} T_{α}T_{α}^{∗}∥

≤ 2 ∥ ∑_{p=1}^{m−1} r^{p} ∑_{|α|=p}(A_{α}^{(k)} − T_{α})(A_{α}^{(k)} − T_{α})^{∗}∥ + ϵ

for any k = 1, 2, . . .. Since A^{(k)} → T in the norm topology, as k → ∞, and using the results above, one can easily deduce that lim_{k→∞} P_{ρ}(rA^{(k)}, R) = P_{ρ}(rT, R) for each r ∈ [0, 1). Now, taking k → ∞ in inequality (5.3), we obtain P_{ρ}(rT, R) ≤ (1 + c^{2}δ)I for r ∈ [0, 1). Applying Theorem 2.2, we deduce

that T ≺^{H} 0. Now, Theorem 2.5 implies r(T) < 1, which shows that T is in [C_{ρ}]^{≺ 0} and, therefore, in [C_{∞}]_{<1}, which proves part (ii).

It remains to prove part (iii). To this end, let {A^{(k)} := (A_{1}^{(k)}, …, A_{n}^{(k)})}_{k=1}^{∞} be a d_{K}-Cauchy sequence in [C_{∞}]_{<1}. Given ϵ > 0, there exists k_{0} ≥ 1 such that d_{K}(A^{(k)}, A^{(j)}) < ϵ for any k, j ≥ k_{0}. Then we have

(5.4)  d_{K}(A^{(k)}, 0) ≤ c := d_{K}(A^{(k_{0})}, 0) + ϵ for any k ≥ k_{0}.

Hence, and due to the definition of d_{K}, we have ∥u(A^{(k)}) − u(0)∥ ≤ c and, consequently,

u(A^{(k)}) ≤ (∥u(A^{(k)} − u(0)∥ + 1)I ≤ (c + 1)u(0) for any k ≥ k_{0}

and for any positive free pluriharmonic function u on [B(H)^{n}]_{1} with coefficients in B(E) such that u(0) = I.

Now, for each k ≥ k_{0}, fix ρ_{k} ≥ 1 such that A^{(k)} ∈ [C_{ρ_{k}}]^{≺ 0}. Note that the inequality above implies

u(A^{(k)}) + (ρ_{k} − 1)u(0) ≤ ρ_{k}(c + 1)u(0)

for all k ≥ k_{0}. Applying Theorem 2.2 and using relation (5.4), we obtain

∥L_{0,A^{(k)}}∥^{2} ≤ d_{K}(A^{(k_{0})}, 0) + ϵ + 1, k ≥ k_{0}.

Consequently, we have

(5.5)  1 ≤ ϵ_{0} := sup_{k≥k_{0}} ∥L_{0,A^{(k)}}∥^{2} < ∞.

Since {A^{(k)}} is a d_{K}-Cauchy sequence, there exists m_{0} ≥ k_{0} such that d_{K}(A^{(m′)}, A^{(m)}) < \frac{1}{2ϵ_{0}} for any m, m′ ≥ m_{0}. Using now relation (5.5), we obtain

(5.6)  d_{K}(A^{(m)}, A^{(m_{0})}) < \frac{1}{2∥L_{0,A^{(m_{0})}}∥^{2}} ,  k ≥ m_{0}.

Since A^{(m_{0})} ∈ [C_{ρ_{m_{0}}} ]^{≺ 0}, Theorem 2.5 implies r(A^{(m_{0})}) < 1. On the other hand, since lim_{ρ→∞} ω_{ρ}(A^{(m_{0})}) = r(A^{(m_{0})}) < 1, there exists ρ_{m_{0}} > 0 such that ω_{ρ_{m_{0}}} (A^{(m_{0})}) < 1 for any ρ ≥ ρ_{m_{0}}. We can assume that

(5.7)  ρ_{m_{0}} ≥ \frac{∥L_{A^{(m_{0})},0}∥^{2}}{∥L_{0,A^{(m_{0})}}∥^{2}} .

Using Proposition 5.2 and relation (5.6), we deduce that

(5.8)  P_{ρ_{m_{0}}} (A^{(m_{0})}, R) ≤ P_{ρ_{m_{0}}} (A^{(k)}, R) + \frac{1}{2∥L_{0,A^{(k)}}∥^{2}} I,  k ≥ m_{0}.

On the other hand, since ω_{ρ_{m_{0}}} (A^{(m_{0})}) < 1, Theorem 3.2 implies A^{(m_{0})} ∼^{H} 0 in C_{ρ_{m_{0}}} . Consequently, we have 0≺^{H} A^{(m_{0})}, which due to Theorem 2.2, implies

ρ_{m_{0}}I = P_{ρ_{m_{0}}} (0, R) ≤ ∥L_{A^{(m_{0})},0}∥^{2}P_{ρ_{m_{0}}} (A^{(m_{0})}, R).

Combining this with relation (5.7), we get

P_{ρ_{m_{0}}} (A^{(m_{0})}, R) ≥ \frac{1}{∥L_{0,A^{(m_{0})}}∥^{2}} I.

Hence, and due to (5.8), we have

P_{ρ_{m_{0}}} (A^{(k)}, R) ≥ \frac{1}{2∥L_{0,A^{(m_{0})}}∥^{2}} I ≥ \frac{1}{2ϵ_{0}} I.

Applying Theorem 3.2, we deduce that A^{(k)} ∼^{H} 0 and A^{(k)} ∈ C_{ρ_{m_{0}}} . Therefore, A^{(k)} ∈ [C_{ρ_{m_{0}}} ]^{≺ 0} for all k ≥ m_{0} and the sequence {A^{(k)}}_{k≥m_{0}} is a d_{K}-Cauchy sequence in [C_{ρ_{m_{0}}} ]^{≺ 0}. Due to part (ii), there exists A ∈ [C_{ρ_{m_{0}}} ]^{≺ 0} ⊂ [C_{∞}]_{<1} such that d_{K}(A^{(k)}, A) → 0, as k → ∞, which proves that d_{K} is a complete metric on [C_{∞}]_{<1}. The proof is complete. □

We can provide now a class of Lipschitz functions with respect to the Carathéodory metric on [C_{∞}]_{<1}.

Theorem 5.4. Let f := (f_{1}, …, f_{m}) be a contractive free holomorphic function with ∥f(0)∥ < 1 such that the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n}. Then

d_{K}(f(A), f(B)) ≤ \frac{1 + ∥f(0)∥}{1 − ∥f(0)∥}d_{K}(A, B)

for any n-tuples A := (A_{1}, …, A_{n}) and B := (B_{1}, …, B_{n}) in [C_{∞}]_{<1}.

Proof. According to the maximum principle for free holomorphic functions with operator-valued coefficients (see Proposition 5.2 from [50]), the condition ∥f(0)∥ < 1 implies that ∥f(X)∥ < 1, X ∈ [B(H)^{n}]_{1}. If u is a free pluriharmonic function on [B(H)^{m}]_{1}, then Theorem 1.1 from [51] shows that u ∘ f is a free pluriharmonic function on [B(H)^{n}]_{1}. If, in addition, u is positive, then u ∘ f is also positive. Assume now that A and B are in [C_{∞}]_{<1}. Due to Theorem 4.10, f(A) and f(B) are in [C_{∞}]_{<1}. Let p ∈ ℂ[X_{1}, …, X_{m}] ⊗ M_{k}, k ∈ ℕ, be a matrix-valued noncommutative polynomial with ℜp(0) = I and ℜp ≥ 0 on [B(H)^{m}]_{1}. According to the Harnack type inequality (4.3), we have

\frac{1 − ∥f(0)∥}{1 + ∥f(0)∥}I ≤ ℜp(f(0)) ≤ \frac{1 + ∥f(0)∥}{1 − ∥f(0)∥}I.

Since ∥f(0)∥ < 1, we deduce that ℜp(f(0)) is a positive invertible operator of the form I_{H} ⊗ A for some A ∈ M_{k}. Define the mapping h : [B(H)^{n}]_{1} → B(H) ¯⊗_{min}M_{k} by setting

h(X) := [ℜp(f(0))]^{−1/2}ℜp(f(X))[ℜp(f(0))]^{−1/2}, X ∈ [B(H)^{n}]_{1}.

Note that h is a positive free pluriharmonic function on [B(H)^{n}]_{1} with coefficients in M_{k} with the property that h(0) = I. Now, using the above-mentioned Harnack type inequality, we have

∥ℜp(f(A)) − ℜp(f(B))∥

≤ ∥[ℜp(f(0))]^{1/2}∥ ∥[ℜp(f(0))]^{−1/2} (ℜp(f(A)) − ℜp(f(B))) [ℜp(f(0))]^{1/2}∥

· ∥[ℜp(f(0))]^{1/2}∥

≤ ∥[ℜp(f(0))]∥∥h(A) − h(B)∥

\frac{1 + ∥f(0)∥}{1 − ∥f(0)∥}d_{K}(A, B).

Taking the supremum over all polynomials p ∈ ℂ[X_{1}, …, X_{m}] ⊗ M_{k}, k ∈ ℕ, with ℜp(0) = I and ℜp ≥ 0 on [B(H)^{m}]_{1}, we obtain

d_{K}(f(A), f(B)) ≤ \frac{1 + ∥f(0)∥}{1 − ∥f(0)∥}d_{K}(A, B),

which completes the proof. □

Corollary 5.5. Let f := (f_{1}, …, f_{m}) be a contractive free holomorphic function with f(0) = 0 such that the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n}. Then

d_{K}(f(A), f(B)) ≤ d_{K}(A, B)

for any A, B ∈ [C_{∞}]_{<1}.

We remark that, using Corollary 1.2 and the remarks preceding Corollary 2.4, one can easily obtain the following result, which provides a simple example when the inequality of Theorem 5.4 is an equality.

Corollary 5.6. If 1 ≤ m < n, let A := (A_{1}, …, A_{m}) ∈ B(H)^{m} and B := (B_{1}, …, B_{m}) ∈ B(H)^{m} be in [C_{∞}]_{<1} and let Ã := (A_{1}, …, A_{m}, 0, …, 0) and B̃ := (B_{1}, …, B_{m}, 0, …, 0) be their extensions in B(H)^{n}, respectively. Then

d_{K}(A, B) = d_{K}(Ã, B̃).

According to Theorem 5.3, the d_{K}-topology coincides with the norm topology on [C_{∞}]_{<1}. Due to Theorem 5.4, we deduce the following result.

Corollary 5.7. Let f := (f_{1}, …, f_{m}) be a contractive free holomorphic function with ∥f(0)∥ < 1 such that the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n}. Then the map

[C_{∞}]_{<1} ∋ (T_{1}, …, T_{n}) ↦ f(T_{1}, …, T_{n}) ∈ [C_{∞}]_{<1}

is continuous in the operator norm topology, where [C_{∞}]_{<1} is the corresponding ball in B(H)^{n} and B(H)^{m}, respectively.

6. Three metric topologies on Harnack parts of C_{ρ}

In this section we study the relation between the δ_{ρ}-topology, the d_{K}-topology, and the operator norm topology on Harnack parts of C_{ρ}. We prove that the hyperbolic metric δ_{ρ} is a complete metric on certain Harnack parts of C_{ρ}, and that all the three topologies coincide on [C_{ρ}]_{<1}. In particular, we prove that the hyperbolic metric δ_{ρ} is complete on the open unit unit ball [C_{ρ}]_{<1}, while the other two metrics are not complete. First, we mention another formula for the hyperbolic distance that will be used to prove the main result of this section. If f ∈ A_{n} ¯⊗_{min}M_{m}, m ∈ ℕ, then we call ℜf strictly positive and denote ℜf > 0 if there exists a constant a > 0 such that ℜ f ≥ aI. We remark that, in this case, if (T_{1}, …, T_{n}) ∈ C_{ρ}, then, using the functional calculus for the class C_{ρ}, we deduce that

ℜf(T_{1}, …, T_{n}) + (ρ − 1)ℜf(0) ≥ ρaI.

The proof of the next result is similar to that of Proposition 3.5 from [49], but uses the functional calculus for the class C_{ρ} and Theorem 2.2 of the present paper. We shall omit it.

Proposition 6.1. Let A := (A_{1}, …, A_{n}) and B := (B_{1}, …, B_{n}) be in C_{ρ} such that A ∼^{H} B. Then

(6.1)  δ_{ρ}(A, B) = \frac{1}{2} sup |ln \frac{⟨[ℜf(A_{1}, …, A_{n}) + (ρ − 1)ℜf(0)]x, x⟩}{⟨[ℜf(B_{1}, …, B_{n}) + (ρ − 1)ℜf(0)]x, x⟩}| ,

where the supremum is taken over all f ∈ A_{n} ⊗ M_{m}, m ∈ ℕ, with ℜf > 0 and x ∈ H ⊗ ℂ^{m} with x ≠ 0.

We remark that, under the conditions of Proposition 6.1, one can also prove that relation (6.1) holds if the supremum is taken over all noncommutative polynomials f ∈ ℂ[X_{1}, …, X_{n}] ⊗ M_{m}, m ∈ ℕ, with ℜf > 0, and x ∈ H ⊗ ℂ^{m} with x ≠ 0. The main result of this section is the following.

Theorem 6.2. Let δ_{ρ}, ρ > 0, be the hyperbolic metric on a Harnack part Δ of [C_{ρ}]^{≺ 0}. Then the following properties hold:

(i) δ_{ρ} is complete on Δ; (ii) the δ_{ρ}-topology is stronger then the d_{K}-topology on Δ; (iii) the δ_{ρ}-topology, the d_{K}-topology, and the operator norm topology coin-

cide on [C_{ρ}]_{<1}; (iv) [C_{ρ}]_{<1} is complete relative the hyperbolic metric, but not complete with

respect to the Carathéodory metric d_{K} and the operator metric.

Proof. Let A := (A_{1}, …, A_{n}) and B := (B_{1}, …, B_{n}) be n-tuples in a Harnack part Δ of [C_{ρ}]^{≺ 0}. Then A is Harnack equivalent to B and

ℜf(A_{1}, …, A_{n}) + (ρ − 1)ℜf(0) ≤ Λ_{ρ}(A, B)^{2}[ℜf(B_{1}, …, B_{n}) + (ρ − 1)ℜf(0)]

for any f ∈ A_{n} ¯⊗_{min}M_{m} with ℜf ≥ 0, where Λ_{ρ}(A, B) is defined by (3.3). Hence, we deduce that

(6.2) ℜf(A_{1}, …, A_{n}) − ℜf(B_{1}, …, B_{n}) ≤ [Λ_{ρ}(A, B)^{2} − 1][ℜf(B_{1}, …, B_{n}) + (ρ − 1)ℜf(0)].

Since B≺^{H} 0, we have the joint spectral radius r(B) < 1, so the ρ-pluriharmonic kernel P_{ρ}(B, R) makes sense. Due to the fact that the noncommutative Poisson transform id ⊗ P_{rR} is completely positive, and P_{ρ}(B, S) ≤ ∥P_{ρ}(B, R)∥I, one can easily see that

\begin{aligned}P_{ρ}(rB, R) & = (id ⊗ P_{rR})[P_{ρ}(B, S)] ≤ ∥P_{ρ}(B, R)∥I \\ & = \frac{1}{ρ}∥P_{ρ}(B, R)∥P_{ρ}(0, R) \end{aligned}

for any r ∈ [0, 1). Using the equivalence (ii) ↔ (iii) of Theorem 2.2, when c^{2} = \frac{1}{ρ}∥P_{ρ}(B, R)∥, we obtain ℜf(rB_{1}, …, rB_{n}) + (ρ − 1)ℜf(0) ≤ ∥P_{ρ}(B, R)∥ℜf(0) for any r ∈ [0, 1). Letting r → 1, in the operator norm topology, we deduce that

ℜf(B_{1}, …, B_{n}) + (ρ − 1)ℜf(0) ≤ ∥P_{ρ}(B, R)∥ℜf(0).

Hence, and using relation (6.2), we obtain

ℜf(A_{1}, …, A_{n}) − ℜf(B_{1}, …, B_{n}) ≤ [Λ_{ρ}(A, B)^{2} − 1]∥P_{ρ}(B, R)∥ℜf(0).

We can obtain a similar inequality if we interchange A with B. If, in addition, we assume that ℜf(0) = I, then we obtain

−tI ≤ ℜf(A_{1}, …, A_{n}) − ℜf(B_{1}, …, B_{n}) ≤ tI,

where t := [Λ_{ρ}(A, B)^{2} − 1] max{∥P_{ρ}(A, R)∥, ∥P_{ρ}(B, R)∥}. On the other hand, since ℜf(A_{1}, …, A_{n}) − ℜf(B_{1}, …, B_{n}) is a self-adjoint operator, we get ∥ℜf(A_{1}, …, A_{n})−ℜf(B_{1}, …, B_{n})∥ ≤ t. Hence, we deduce that d_{K}(A, B) ≤ s. As a consequence, we obtain

(6.3)  d_{K}(A, B) ≤ max{∥P_{ρ}(A, R)∥, ∥P_{ρ}(B, R)∥} (e^{2δ_{ρ}(A,B)} − 1) .

Let us prove that δ_{ρ} is a complete metric on Δ. To this end, let {A^{(k)} := (A_{1}^{(k)}, …, A_{n}^{(k)})}_{k=1}^{∞} ⊂ Δ be a δ_{ρ}-Cauchy sequence. First, we prove that the sequence {∥P_{ρ}(A^{(k)}, R)∥}_{k=1}^{∞} is bounded. Given ϵ > 0, there exists k_{0} ∈ ℕ such that

(6.4)  δ_{ρ}(A^{(k)}, A^{(p)}) < ϵ for any k, p ≥ k_{0}.

Let f ∈ A_{n} ¯⊗_{min}M_{m} with Re f ≥ 0. Since A^{(k_{0})}≺^{H} 0 and

P_{ρ}(rA^{(k_{0})}, R) ≤ \frac{1}{ρ}∥P_{ρ}(rA^{(k_{0})}, R)∥P_{ρ}(0, R),

Theorem 2.2 implies

ℜf(A^{(k_{0})}) + (ρ − 1)ℜf(0) ≤ \frac{1}{ρ}∥P_{ρ}(rA^{(k_{0})}, R)∥[ℜf(0) + (ρ − 1)ℜf(0)].

On the other hand, since A^{(k)} ∼^{H} A^{(k_{0})}, Theorem 2.2 implies

ℜf(A^{(k)}) + (ρ − 1)ℜf(0) ≤ Λ_{ρ}(A^{(k)}, A^{(k_{0})})^{2}[ℜf(A^{(k_{0})}) + (ρ − 1)ℜf(0)].

Combining these inequalities, we obtain

\begin{aligned}& ℜf(A^{(k)}) + (ρ − 1)ℜf(0) ≤ c^{2} \frac{1}{ρ}[ℜf(0) + (ρ − 1)ℜf(0)], \\ (6.5) \end{aligned}

where c := ∥P_{ρ}(A^{(k_{0})}, R)∥^{1/2}Λ_{ρ}(A^{(k)}, A^{(k_{0})}), for any f ∈ A_{n}⊗M_{m} with ℜf ≥ 0. Consequently, due to Theorem 2.2, we have ∥P_{ρ}(A^{(k)}, R)∥ ≤ c^{2} for any k ≥ k_{0}. Combining this with relation (6.4), we obtain

∥P_{ρ}(A^{(k)}, R)∥ ≤ ∥P_{ρ}(A^{(k_{0})}, R)∥e^{2ϵ}

for any k ≥ k_{0}. This shows that the sequence {∥P_{ρ}(A^{(k)}, R)∥}_{k=1}^{∞} is bounded. Consequently, inequality (6.3) implies that {A^{(k)}} is a d_{K}-Cauchy sequence. Due to Theorem 5.3, there exists A := (A_{1}, …, A_{n}) ∈ [C_{ρ}]^{≺ 0} such that

(6.6)  d_{K}(A^{(k)}, A) → 0 as k → ∞.

In what follows, we prove that A ∈ Δ. Let f ∈ A_{n} ⊗ M_{m} with ℜf ≥ 0 and ℜf(0) = I. Taking into account relations (6.5) and (6.4), we have

\begin{aligned}ℜf(A^{(k)}) + (ρ − 1)ℜf(0) & ≤ Λ_{ρ}(A^{(k)}, A^{(k_{0})})^{2}[ℜf(A^{(k_{0})}) + (ρ − 1)ℜf(0)] \\ & ≤ e^{2ϵ}[ℜf(A^{(k_{0})}) + (ρ − 1)ℜf(0)] \end{aligned} ; (6.7)

for k ≥ k_{0}. According to relation (6.6) and the definition of d_{K}, ℜf(A^{(k)}) → ℜf(A), as k → ∞, in the operator norm topology. Consequently, relation (6.7) implies

(6.8)  ℜf(A) + (ρ − 1)ℜf(0) ≤ e^{2ϵ}[ℜf(A^{(k_{0})}) + (ρ − 1)ℜf(0)].

Such an inequality can be deduced in the more general case when f ∈ A_{n} ⊗M_{m} with ℜf ≥ 0. Indeed, for each ϵ′ > 0 let g := ϵ′I + f, Y := ℜg(0), and φ := Y^{−1/2}gY^{−1/2}. Since ℜφ ≥ 0 and ℜφ(0) = I, we can apply inequality (6.8) to φ and deduce that

ρϵ′I + ℜf(A) + (ρ − 1)ℜf(0) ≤ e^{2ϵ} [ρϵ′I + ℜf(A^{(k_{0})}) + (ρ − 1)ℜf(0)]

for any ϵ′ > 0. Letting ϵ′ → 0, we get

(6.9)  ℜf(A) + (ρ − 1)ℜf(0) ≤ e^{2ϵ}[ℜf(A^{(k_{0})}) + (ρ − 1)ℜf(0)]

for any f ∈ A_{n} ⊗ M_{m} with ℜf ≥ 0. Therefore,

(6.10)  A≺^{H} A^{(k_{0})}.

On the other hand, since A^{(k_{0})}≺^{H} A^{(k)} for any k ≥ k_{0}, Theorem 2.2 and relation (6.4), imply

\begin{aligned}ℜp(A^{(k_{0})}) + (ρ − 1)ℜp(0) & ≤ Λ_{ρ}(A^{(k_{0})}, A^{(k)})^{2}[ℜp(A^{(k)}) + (ρ − 1)ℜp(0)] \\ & ≤ e^{2ϵ}[ℜp(A^{(k)}) + (ρ − 1)ℜ(0)] \end{aligned}

for k ≥ k_{0} and any polynomial p ∈ ℂ[X_{1}, …, X_{n}] ⊗ M_{m}, m ∈ ℕ, with ℜp ≥ 0. According to Theorem 5.3, the d_{K}-topology coincides with the norm topology on [C_{ρ}]^{≺ 0}. Therefore, relation (6.6) implies A^{(k)} → A ∈ [C_{ρ}]^{≺ 0} in the operator norm topology. Taking the limit, as k → ∞, in the inequality above, we deduce that

(6.11)  ℜp(A^{(k_{0})}) + (ρ − 1)ℜp(0) ≤ e^{2ϵ}[ℜp(A) + (ρ − 1)ℜp(0)]

for any p ∈ ℂ[X_{1}, …, X_{n}] ⊗ M_{m} with ℜp ≥ 0. Consequently, we get A^{(k_{0})}≺^{H} A. Hence, and using relation (6.10), we obtain A∼^{H} A^{(k_{0})}, which proves that A ∈ Δ. The inequalities (6.9) and (6.11) imply Λ_{ρ}(A^{(k_{0})}, A) ≤ e^{2ϵ}. This shows that δ_{ρ}(A^{(k_{0})}, A) < ϵ, which together with relation (6.4) imply δ_{ρ}(A^{(k)}, A) < 2ϵ for any k ≥ k_{0}. Therefore, δ_{ρ}(A^{(k)}, A) → 0, as k → ∞, which proves that δ_{ρ} is a complete metric on the Harnack part Δ. Note that we have also proved part (ii) of this theorem.

In what follows, we prove part (iii). To this end, assume that A and B are n-tuples of operators in [C_{ρ}]_{<1}. Due to Theorem 3.2, P_{ρ}(B, R) is a positive invertible operator. Since P_{ρ}(B, R)^{−1} ≤ ∥P_{ρ}(B, R)^{−1}∥, we have I ≤ ∥P_{ρ}(B, R)^{−1}∥P_{ρ}(B, R), which, applying the noncommutative Poisson transform, implies I ≤ ∥P_{ρ}(B, R)^{−1}∥P_{ρ}(rB, R) for any r ∈ [0, 1). By The-

orem 2.2, we deduce that 0≺^{H} B and

ℜf(0) ≤ ∥P_{ρ}(B, R)^{−1}∥ [ℜf(B) + (ρ − 1)ℜf(0)]

for any f ∈ A_{n} ⊗ M_{m} with ℜf ≥ 0. If, in addition, ℜf(0) = I, then the latter inequality implies

\begin{aligned}\frac{⟨[ℜf(A) + (ρ − 1)ℜf(0)]x, x⟩}{⟨[ℜf(B) + (ρ − 1)ℜf(0)]x, x⟩} − 1 & ≤ \frac{∥P_{ρ}(B, R)^{−1}∥}{∥x∥} ⟨(ℜf(A) − ℜf(B)) x, x⟩ \\ & ≤ ∥P_{ρ}(B, R)^{−1}∥d_{K}(A, B) \end{aligned}

for any x ∈ H ⊗ ℂ^{m}, x ≠ 0. Consequently, we have

ln \frac{⟨[ℜf(A) + (ρ − 1)ℜf(0)]x, x⟩}{⟨[ℜf(B) + (ρ − 1)ℜf(0)]x, x⟩} ≤ ln (1 + ∥P_{ρ}(B, R)^{−1}∥d_{K}(A, B)) .

A similar inequality can be obtained interchanging A with B. Combining these two inequalities, we get

\begin{matrix}| & ln & \frac{⟨[ℜf(A) + (ρ − 1)ℜf(0)]x, x⟩}{⟨[ℜf(B) + (ρ − 1)ℜf(0)]x, x⟩} & | \\ ≤ ln (1 + max\{∥P_{ρ}(B, R)^{−1}∥, ∥P_{ρ}(A, R)^{−1}∥\}d_{K}(A, B)) . \end{matrix} ; (6.12)

Now, we consider the general case when g ∈ A_{n} ⊗ M_{m} with ℜg > 0. Note that Y := ℜg(0) is a positive invertible operator on H ⊗ ℂ^{m} and f := Y^{−1/2}gY^{−1/2} has the properties ℜf ≥ 0 and ℜf(0) = I. Applying inequality (6.12) to f when x := Y^{−1/2}y, y ∈ H ⊗ ℂ^{m}, and y ≠ 0, we obtain

(6.13) 2δ_{ρ}(A, B) ≤ ln (1 + max{∥P_{ρ}(B, R)^{−1}∥, ∥P_{ρ}(A, R)^{−1}∥}d_{K}(A, B)) .

Consider a sequence {A^{(k)}}_{k=1}^{∞} of elements in [C_{ρ}]_{<1} and let A ∈ [C_{ρ}]_{<1} be such that d_{K}(A^{(k)}, A) → 0, as k → ∞. By Proposition 5.2, we deduce that P_{ρ}(A^{(k)}, R) → P_{ρ}(A, R) in the operator norm topology. On the other hand, due to Theorem 3.2, the operators P (A^{(k)}, R) and P (A, R) are invertible. Hence, and using the well-known fact that the map Z ↦ Z^{−1} is continuous on the open set of all invertible operators, we deduce that P_{ρ}(A^{(k)}, R)^{−1} → P_{ρ}(A, R)^{−1} in the operator norm topology, as k → ∞. Hence, we deduce that the sequence {∥P_{ρ}(A^{(k)}, R)^{−1}∥}_{k=1}^{∞} is bounded. Consequently, there exists M > 0 with ∥P_{ρ}(A^{(k)}, R)^{−1}∥ ≤ M for any k ∈ ℕ. Using inequality (6.13), we obtain

2δ_{ρ}(A^{(k)}, A) ≤ ln (1 + Md_{K}(A^{(k)}, A)) , k ∈ ℕ.

Since d_{K}(A^{(k)}, A) → 0, as k → ∞, the latter inequality implies that δ_{ρ}(A^{(k)}, A) → 0. Therefore, the d_{K}-topology on [C_{ρ}]_{<1} is stronger than the δ_{ρ}-topology. Due to the first part of this theorem, the two topologies coincide on [C_{ρ}]_{<1}. Using now Theorem 5.3, we complete the proof of part (iii). Now, we prove item (iv). Since [C_{ρ}]_{<1} is the Harnack part of 0 (see Theorem 3.2), part (i) implies its completeness with respect to the hyperbolic metric. To prove that [C_{ρ}]_{<1} is not complete with respect to the Carathéodory metric d_{K} and the operator metric, we consider the following example. Let (T_{1}, …, T_{n}) ∈ B(P_{1})^{n} be the n-tuple of operators defined by T_{i} := P_{P_{1}}S_{i}|_{P_{1}}, i = 1, …, n, where P_{1} := span{e_{α} : |α| ≤ 1}. Note that ∥[T_{1}, …, T_{n}]∥ = 1 and T_{α} = 0 for any α ∈ 𝔽_{n}^{+} with |α| ≥ 2. Set X_{i} := ρT_{i}, i = 1, …, n, and note that

X_{β} = ρT_{β} = ρP_{P_{1}}S_{β}|_{P_{1}}, β ∈ 𝔽_{n}^{+}\{g_{0}}.

Therefore, (X_{1}, …, X_{n}) ∈ C_{ρ}, i.e., ω_{ρ}(X_{1}, …, X_{n}) ≤ 1, which implies ω_{ρ}(T_{1}, …, T_{n}) ≤ \frac{1}{ρ}. The reverse inequality is due to the fact that ∥[T_{1}, …, T_{n}]∥ ≤ ρω_{ρ}(T_{1}, …, T_{n}). Consequently, we have

ω_{ρ}(T_{1}, …, T_{n}) = \frac{1}{ρ}, for ρ ∈ (0, ∞).

On other hand, the condition T_{α} = 0 if |α| ≥ 2 implies r(T_{1}, …, T_{n}) = 0. Therefore, we have

ω_{ρ}(X_{1}, …, X_{n}) = 1 and r(X_{1}, …, X_{n}) = 0.

Now, let c ∈ (0, 1) and define Y^{(k)} := c^{1/k}(X_{1}, …, X_{n}) for k = 1, 2, …. Since ω_{ρ}(Y^{(k)}) = c^{1/n} < 1, Theorem 3.2 implies Y^{(k)} ∼^{H} 0 in C_{ρ} and Y^{(k)} ∈ [C_{ρ}]_{<1}. On the other hand, since ω_{ρ}(X_{1}, …, X_{n}) = 1, we have X := (X_{1}, …, X_{n}) ∉ [C_{ρ}]_{<1}. Now, note that

\begin{aligned}d_{K}(Y^{(k)}, X) & ≤ 2∥(I − R_{Y^{(k)}})^{−1} − (I − R_{X})^{−1}∥ \\ & = 2∥R_{Y^{(k)}} − R_{X}∥ = 2∥Y^{(k)} − X∥ = 2∥X∥(1 − c^{1/k}). \end{aligned}

Consequently, Y^{(k)} → X in the operator norm and d_{K}(Y^{(k)}, X) → 0, as k → ∞. This shows that [C_{ρ}]_{<1} is not complete with respect to the Carathéodory metric d_{K} and the operator metric. The proof is complete. □

Corollary 6.3. Let δ_{ρ} be the hyperbolic metric on a Harnack part Δ of [C_{ρ}]^{≺ 0}. Then

d_{K}(A, B) ≤ max{∥P_{ρ}(A, R)∥, ∥P_{ρ}(B, R)∥} (e^{2δ_{ρ}(A,B)} − 1) , A, B ∈ Δ.

If, in addition A, B ∈ [C_{ρ}]_{<1}, then

2δ_{ρ}(A, B) ≤ ln (1 + max{∥P_{ρ}(B, R)^{−1}∥, ∥P_{ρ}(A, R)^{−1}∥}d_{K}(A, B)) .

Corollary 6.4. Let f := (f_{1}, …, f_{m}) be a contractive free holomorphic function with ∥f(0)∥ < 1 such that the boundary functions f̃_{1}, …, f̃_{m} are in the noncommutative disc algebra A_{n}. If Δ is a Harnack part of [C_{ρ}]^{≺ 0}, then the map

Δ ∋ (T_{1}, …, T_{n}) ↦ f(T_{1}, …, T_{n}) ∈ [C_{ρ_{f}}]^{≺ 0} is continuous with respect to the hyperbolic metric δ_{ρ} on Δ and the Carathéodory metric d_{K} on [C_{ρ_{f}}]^{≺ 0}, where ρ_{f} is defined by relation (4.1). In particular, tha map

[C_{ρ}]_{<1} ∋ (T_{1}, …, T_{n}) ↦ f(T_{1}, …, T_{n}) ∈ [C_{ρ_{f}}]_{<1}

is continuous with respect to the hyperbolic metric.

7. Harnack domination and hyperbolic metric for ρ-contractions (case n = 1)

In this section, we consider the single variable case (n = 1) and show that our Harnack domination of ρ-contractions is equivalent to the one introduced and studied by Cassier and Suciu in [9]. We recover some of their results and obtain some results which seem to be new even in the single variable case.

In the particular case when n = 1, the free pluriharmonic Poisson kernel P_{ρ}(rY, R), r ∈ [0, 1), coincides with

Q_{ρ}(rY, U) := ∑_{k=1} r^{k}Y^{∗}^{k} ⊗ U^{k} + ρI ⊗ I + ∑_{k=1}^{∞} r^{k}Y^{k} ⊗ U^{∗}^{k}, Y ∈ C_{ρ} ⊂ B(H),

where the convergence of the series is in the operator norm topology and U is the unilateral shift acting on the Hardy space H^{2}(𝕋). For each ρ-contraction T ∈ B(H), consider the operator-valued Poisson kernel defined by

K_{ρ}(z, T) := ∑_{k=1}^{∞} z^{k}T^{∗}^{k} + ρI + ∑_{k=1}^{∞} z̄^{k}T^{k}, z ∈ 𝔻,

which was employed by Cassier and Fack in [8]. Using Theorem 2.2, in the particular case when n = 1, we can prove the following result.

Proposition 7.1. Let T and T′ be two ρ-contractions in B(H) and let c ≥ 1. Then the following statements are equivalent:

(i) T ≺_{c}^{H} T′;

(ii) Q_{ρ}(rT, U) ≤ c^{2}Q_{ρ}(rT′, U) for any r ∈ [0, 1); (iii) K_{ρ}(z, T) ≤ c^{2}K_{ρ}(z, T′) for any z ∈ 𝔻.

Proof. The equivalence (i) ↔ (ii) follows from Theorem 2.2, when n = 1. To prove the implication (ii) ⟹ (iii), we apply the noncommutative Poisson transform (when n = 1) at e^{it}I to the inequality of part (ii). Consequently, we obtain

\begin{aligned}K_{ρ}(re^{it}, T) & = (id ⊗ P_{e^{it}I})[Q_{ρ}(rT, U)] \\ & ≤ c^{2}(id ⊗ P_{e^{it}I})[Q_{ρ}(rT′, U)] = c^{2}K_{ρ}(re^{it}, T′) \end{aligned}

for any r ∈ [0, 1) and t ∈ ℝ. Now let us prove that (iii) ⟹ (ii). Since

\begin{matrix}⟨(T^{∗}^{k} ⊗ U^{k})(h_{m} ⊗ e^{imt}), h_{p} ⊗ e^{ipt}⟩ & H⊗H^{2}(𝕋) \\ = \frac{1}{2π} ∫_{−π}^{π} ⟨e^{ikt}T^{∗}^{k}(e^{imt}h_{m}), e^{ipt}h_{p}⟩_{H} dt \end{matrix}

for any h_{m}, h_{p} ∈ H and k, m, p ∈ ℕ, one can easily obtain

\begin{matrix}⟨(c^{2}Q_{ρ}(rT′, U) − Q_{ρ}(rT, U)) h(e^{it}), h(e^{it})⟩ & H⊗H^{2}(𝕋) \\ = \frac{1}{2π} ∫_{−π}^{π} ⟨(c^{2}K_{ρ}(re^{it}, T′) − K_{ρ}(re^{it}, T)) h(e^{it}), h(e^{it})⟩_{H} \end{matrix}

for any function e^{it} ↦ h(e^{it}) in H⊗H^{2}(𝕋). Now, the implication (iii) ⟹ (ii) is clear. The proof is complete. □

Let T, T′ ∈ B(H) be ρ-contractions such that T ≺^{H} T′. Due to Proposition 7.1 and Corollary 2.3, we deduce that

∥L_{T′,T} ∥ = inf{c > 1 : Q_{ρ}(rT, U) ≤ c^{2}Q_{ρ}(rT′, U) for any r ∈ [0, 1)}
= inf{c > 1 : K_{ρ}(z, T) ≤ c^{2}K_{ρ}(z, T′) for any z ∈ 𝔻}

= inf{c > 1 : K_{ρ}(z, T^{∗}) ≤ c^{2}K_{ρ}(z, T′^{∗}) for any z ∈ 𝔻} = ∥L_{T′^{∗},T^{∗}}∥.

Therefore T ≺^{H} T′ if and only if T^{∗} ≺^{H} T′^{∗}.

Theorem 7.2. Let T, T′ ∈ B(H) be such that T, T′ ∈ [C_{ρ}]_{<1}. Then

∥L_{T′,T} ∥ = sup_{z∈𝔻} ∥Δ_{ρ,T′^{∗}}(z)^{−1}(I − z̄T′^{∗})(I − z̄T^{∗})^{−1}Δ_{ρ,T^{∗}}(z)∥,

where

Δ_{ρ,T} (z) := [ρI + (1 − ρ)(zT^{∗} + z̄T) + (ρ − 2)T T^{∗}]^{1/2}, z ∈ 𝔻.

Moreover,

δ_{ρ}(T, T′) = ln max {∥L_{T,T′}∥ , ∥L_{T′,T} ∥} .

Proof. If T, T′ ∈ [C_{ρ}]_{<1}, Theorem 3.4 implies

∥L_{T′,T} ∥ = ∥L_{T′^{∗},T^{∗}}∥ = sup_{z∈𝔻} ∥Δ_{ρ,T^{∗}}(z)(I − zT)^{−1}(I − zT′)Δ_{ρ,T′^{∗}}(z)^{−1}∥

= sup_{z∈𝔻} ∥Δ_{ρ,T′^{∗}}(z)^{−1}(I − z̄T′^{∗})(I − z̄T^{∗})^{−1}Δ_{ρ,T^{∗}}(z)∥.

Using now Theorem 3.5, we complete the proof. □

We mention that when ρ = 1, we recover a result obtained by I. Suciu [53], using different methods. However, if ρ > 0 and ρ ≠ 1, the result of Theorem 7.2 seems to be new. We also remark that Proposition 3.12 , Proposition 5.2, and part (i) of Theorem 5.3 are new even in the single variable case (n = 1).

The next result makes an interesting connection between the Harnack domination for n-tuples of operators in C_{ρ} and and the Harnack domination for ρ-contractions (n = 1), via the reconstruction operator.

Theorem 7.3. Let A := (A_{1}, …, A_{n}) and B := (B_{1}, …, B_{n}) be in C_{ρ} and let c > 0. Then the following statements are equivalent:

(i) A≺_{c}^{H} B;

(ii) R_{A}≺_{c}^{H} R_{B}, where R_{X} := X_{1}^{∗} ⊗ R_{1} + ⋯ + X_{n}^{∗} ⊗ R_{n} is the reconstruction

operator associated with X := (X_{1}, …, X_{n}) ∈ C_{ρ} and the right creation operators R_{1}, …, R_{n}.

(iii) R_{A}^{∗}≺_{c}^{H} R_{B}^{∗}.

Proof. First, assume that item (i) holds. Due to Theorem 2.2, we have

(7.1)  P_{ρ}(rA, S) ≤ c^{2}P_{ρ}(rB, S)

for any r ∈ [0, 1), where S := (S_{1}, …, S_{n}) is the n-tuple of left creation operators. Let U be the unilateral shift on the Hardy space H^{2}(𝕋). Since R_{i}^{∗}R_{j} = δ_{ij}I, the n-tuple (R_{1} ⊗ U^{∗}, …, R_{n} ⊗ U^{∗}) is a row contraction acting from [F^{2}(H_{n}) ⊗ H^{2}(𝕋)]^{n} to F^{2}(H_{n}) ⊗ H^{2}(𝕋). Applying the noncommutative Poisson transform at (R_{1} ⊗ U^{∗}, …, R_{n} ⊗ U^{∗}) to inequality (7.1), we obtain

\begin{aligned}Q_{ρ}(rR_{A}, U) & = (id ⊗ P_{(R_{1}⊗U^{∗},...,R_{n}⊗U^{∗})}) [P_{ρ}(rA, S)] \\ & ≤ c^{2} (id ⊗ P_{(R_{1}⊗U^{∗},...,R_{n}⊗U^{∗})}) [P_{ρ}(rB, S)] = c^{2}Q_{ρ}(rR_{B}, U) \end{aligned}

for any r ∈ [0, 1). Using Proposition 7.1, we obtain that R_{A}≺_{c}^{H} R_{B}. Now, assume

that (ii) holds. Proposition 7.1 implies

(7.2)  K_{ρ}(re^{it}, R_{A}) ≤ c^{2}K_{ρ}(re^{it}, R_{B}),  r ∈ [0, 1) and t ∈ ℝ.

Taking t = 0, we obtain P_{ρ}(rA, R) ≤ c^{2}P_{ρ}(rB, R) for any r ∈ [0, 1), which, due

to Theorem 2.2, implies A≺_{c}^{H} B. The equivalence (ii) ↔ (iii) is a consequence

of Proposition 7.1 and the fact that inequality (7.2) is equivalent to

K_{ρ}(re^{it}, R_{A}^{∗}) ≤ c^{2}K_{ρ}(re^{it}, R_{B}^{∗}), r ∈ [0, 1) and t ∈ ℝ.

This completes the proof. □

We remark that, according to Theorem 3.4 and Corollary 2.3, we have

∥L_{B,A}∥ = ∥C_{ρ,A}C_{ρ,B}^{−1}∥ = inf{c > 1 : P_{ρ}(A, R) ≤ c^{2}P_{ρ}(B, R)}

for any A, B ∈ [C_{ρ}]_{<1}, where C_{ρ,A} is defined in Theorem 3.4.

Corollary 7.4. If A, B are n-tuples of operators in [C_{ρ}]_{<1}, then ∥L_{B,A}∥ = ∥L_{R_{B},R_{A}}∥ = ∥L_{R_{B}^{∗},R_{A}^{∗}}∥. Moreover, δ_{ρ}(A, B) = δ_{ρ}(R_{A}, R_{B}).

References

[1] T. Ando, I. Suciu, D. Timotin, Characterization of some Harnack parts of contractions, J. Operator Theory 2 (1979), no. 2, 233–245.

[2] C. Badea and G. Cassier, Constrained von Neumann inequalities, Adv. Math. 166 (2002), 260–297.

[3] C.A. Berger, A strange dilation theorem, Abstract 625–152, Notices Amer. Math. Soc. 12 (1965), 590.

[4] C.A. Berger and J.G. Stampfli, Mappings theorems for numerical ranges, Amer J. Math. 89 (1967), 1047–1055.

[5] C.A. Berger and J.G. Stampfli, Norm relations and skew dilations, Acta Sci. Math. (Szeged) 28 (1967), 191–195.

[6] S. Bergman, The kernel function and conformal mapping, Mathematical Surveys, No. V. American Mathematical Society, Providence, R.I., 1970. x+257 pp.

[7] J. W. Bunce, Models for n-tuples of noncommuting operators, J. Funct. Anal. 57(1984), 21–30.

[8] G. Cassier and T. Fack, Un noyau pour divers calculs fonctionnels, C. R. Acad. Sci. Paris Sr. I Math. 317 (1993), no. 7, 683–688.

[9] G. Cassier and N. Suciu, Mapping theorems and Harnack ordering for ρ-contractions, Indiana Univ. Math. J. 55 (2006), no. 2, 483–523.

[10] G. Cassier and N. Suciu, Analytic functions of a uniformly stable ρcontraction. Operator theory 20, 55–73, Theta Ser. Adv. Math., 6, Theta, Bucharest, 2006.

[11] J. Cuntz, Simple C^{∗}–algebras generated by isometries, Commun. Math. Phys. 57 (1977), 173–185.

[12] K. R. Davidson Free semigroup algebras. A survey. Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000), 209–240, Oper. Theory Adv. Appl., 129, Birkhuser, Basel, 2001.

[13] E.G. Effros and Z.J. Ruan, Operator spaces, London Mathematical Society Monographs. New Series, 23. The Clarendon Press, Oxford University Press, New York, 2000.

[14] K. Fan, Analytic functions of a proper contraction, Math. Z 160 (1978), 275–290.

[15] C. Foiaş, On Harnack parts of contractions. Rev. Roumaine Math. Pures Appl. 19 (1974), 315–318.

[16] A. E. Frazho, Models for noncommuting operators, J. Funct. Anal. 48 (1982), 1–11.

[17] P.R. Halmos, A Hilbert space problem book, D. Van Nostrand, Princeton, NJ, 1967.

[18] P.R. Halmos, Positive definite sequences and the miracle of w, 17 pages, mimeographed.

[19] J.A.R. Holbrook, On the power-bounded operators of Sz.-Nagy and Foias, Acta. Sci. Math. (Szeged) 29 (1968), 299-310.

[20] J.A.R. Holbrook, Multiplicative propreties of the numerical radius in operator theory, J. Reine Angew. Math. 237 (1969), 166–174.

[21] T. Kato, Some mapping theorems for the numerical range, Proc. Japan Acad. 41 (1965), 652-655.

[22] S. Kobayashi, Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],

318. Springer-Verlag, Berlin, 1998. xiv+471 pp.

[23] S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. xii+148 pp.

[24] S.G. Krantz, Geometric function theory, Explorations in complex analysis. Cornerstones. Birkhuser Boston, Inc., Boston, MA, 2006. xiv+314 pp.

[25] V. Müller, On the joint spectral radius, Ann. Polon. Math. 66 (1997), 173–182.

[26] P.S. Muhly and B. Solel, Tensor algebras over C^{∗}-correspondences: representations, dilations, and C^{∗}-envelopes, J. Funct. Anal. 158 (1998), 389–457.

[27] P.S. Muhly and B. Solel, Hardy algebras, W^{∗}-correspondences and interpolation theory, Math. Ann. 330 (2004), 353–415.

[28] P.S. Muhly and B. Solel, Schur Class Operator Functions and Automorphisms of Hardy Algebras, Documenta Math. 13 (2008), 365–411.

[29] V.I. Paulsen, Completely Bounded Maps and Dilations, Pitman Research Notes in Mathematics, Vol.146, New York, 1986.

[30] C. Pearcy, An elementary proof of the power inequality for the numerical range, Michigan Math. J. 13 (1966), 289–291.

[31] G. Pisier, Similarity Problems and Completely Bounded Maps, Springer Lect. Notes Math., Vol.1618, Springer-Verlag, New York, 1995.

[32] G. Popescu, Models for infinite sequences of noncommuting operators, Acta. Sci. Math. (Szeged) 53 (1989), 355–368.

[33] G. Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math. Soc. 316 (1989), 523–536.

[34] G. Popescu, Characteristic functions for infinite sequences of noncommuting operators, J. Operator Theory 22 (1989), 51–71.

[35] G. Popescu, Multi-analytic operators and some factorization theorems, Indiana Univ. Math. J. 38 (1989), 693–710.

[36] G. Popescu, Von Neumann inequality for (B(H)^{n})_{1}, Math. Scand. 68 (1991), 292–304.

[37] G. Popescu, Functional calculus for noncommuting operators, Michigan Math. J. 42 (1995), 345–356.

[38] G. Popescu, Multi-analytic operators on Fock spaces, Math. Ann. 303 (1995), 31–46.

[39] G. Popescu, Positive-definite functions on free semigroups, Canad. J. Math. 48 (1996), no. 4, 887–896.

[40] G. Popescu, Noncommutative disc algebras and their representations, Proc. Amer. Math. Soc. 124 (1996), 2137–2148.

[41] G. Popescu, Poisson transforms on some C^{∗}-algebras generated by isometries, J. Funct. Anal. 161 (1999), 27–61.

[42] G. Popescu, Curvature invariant for Hilbert modules over free semigroup algebras, Adv. Math. 158 (2001), 264–309.

[43] G. Popescu, Similarity and ergodic theory of positive linear maps, J. Reine Angew. Math. 561 (2003), 87–129.

[44] G. Popescu, Free holomorphic functions on the unit ball of B(H)^{n}, J. Funct. Anal. 241 (2006), 268–333.

[45] G. Popescu, Characteristic functions and joint invariant suspaces, J. Funct. Anal. 237 (2006), 277–320.

[46] G. Popescu, Hyperbolic geometry on the unit ball of B(H)^{n} and dilation theory, Indiana Univ. Math. J. 57 (2008), No.6, 2891-2930.

[47] G. Popescu, Noncommutative transforms and free pluriharmonic functions, Adv. Math. 220 (2009), 831-893.

[48] G. Popescu, Unitary invariants in multivariable operator theory, Mem. Amer. Math. Soc. 200 (2009), no. 941, vi+91 pp.

[49] G. Popescu, Noncommutative hyperbolic geometry on the unit ball of B(H)^{n}, J. Funct. Anal. 256 (2009), 4030-4070.

[50] G. Popescu, Free holomorphic automorphisms of the unit ball of B(H)^{n}, J. Reine Angew. Math., to appear.

[51] G. Popescu, Free holomorphic functions on the unit ball of B(H)^{n}. II., J. Funct. Anal., to appear.

[52] W. Rudin, Function theory in the unit ball of ℂ^{n}, Springer-verlag, New-York/Berlin, 1980.

[53] I. Suciu, Analytic formulas for the hyperbolic distance between two contractions, Ann. Polon. Math. 66 (1997), 239–252.

[54] B. Sz.-Nagy, Sur les contractions de l’espace de Hilbert, Acta. Sci. Math. (Szeged) 15 (1953), 87–92.

[55] B. Sz.-Nagy and C. Foiaş, On certain classes of power-bounded operators in Hilbert space, Acta. Sci. Math. (Szeged) 27 (1966), 17–25.

[56] B. Sz.-Nagy and C. Foiaş, Harmonic Analysis of Operators on Hilbert Space, North Holland, New York 1970.

[57] J. von Neumann, Eine Spectraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951), 258–281.

[58] K. Zhu, Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics, 226. Springer-Verlag, New York, 2005. x+271 pp.

[59] J.P. Williams, Schwarz norms for operators, Pacific J. Math. 24 (1968), 181–188.

Gelu Popescu Department of Mathematics The University of Texas at San Antonio San Antonio TX 78249 USA gelu.popescu@utsa.edu 652

Documenta Mathematica 14 (2009)