Lobachevskii Journal of Mathematics

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Vol. 18, 2005, 21–32 \begin{matrix}c \\ ◯ \end{matrix} A. L. Barrenechea, C. C. Peña

A. L. Barrenechea and C. C. Peña

ON INNERNESS OF DERIVATIONS ON S(H)

(submitted by D. Kh. Mushtari)

Abstract. We consider general bounded derivations on the Banach algebra of Hilbert-Schmidt operators on an underlying complex infinite dimensional separable Hilbert space H. Their structure is described by means of unique infinite matrices. Certain classes of derivations are identified together in such a way that they correspond to a unique matrix derivation. In particular, Hadamard derivations, the action of general derivations on Hilbert-Schmidt and nuclear operators and questions about innerness are considered.

1. Introduction

Throughout this article H will be a separable infinitely dimensional complex Hilbert space. Let B (H), K (H), S (H) and N (H) be the classes of bounded, compact, Hilbert-Schmidt and nuclear operators on H respectively. As it is well known, K (H) is the only non-trivial closed self-adjoint two-sided ideal in B(H) (cf. [5]). Furthermore, by the polar decomposition theorem any A ∈ B(H) can be written uniquely as A = U ∘ |A|, where U is a partial isometry and |A| is a positive operator (see [8], Vol. 2, Th. 6.1.2, p. 401). Remember that if A ∈ K(H) and {ρ_{n}} is the sequence of eigenvalues of |A| then A is said to be a Hilbert-Schmidt operator (resp. a nuclear operator) if ∑ ρ_{n}^{2} < ∞ (resp. if ∑ ρ_{n} < ∞). Moreover, an operator A is of Hilbert-Schmidt type if and only if the series ∑ ∥Af_{n}∥^{2} converges for at least one orthonormal basis {f_{n}} of H. In that case it is readily seen that ∑ ρ_{n}^{2} = ∑ ∥Ag_{n}∥^{2} if {g_{n}} is any orthonormal basis of H. So, if {e_{n}} is an orthonormal basis of H and A, B ∈ S(H) then

2000 Mathematical Subject Classification. 46H05, 46J45, 47B47.

Key words and phrases. Hilbert-Schmidt and nuclear operator, Nearlyinner matrices, Hadamard products.

⟨A, B⟩_{2} = ∑ ⟨Ae_{n}, Be_{n}⟩

defines an inner product on S(H). If ∥A∥_{2} = ⟨A, A⟩_{2}^{1/2} then (S(H), ⟨∘, ∘⟩_{2}) becomes a Hilbert space. Indeed, (S(H), ∥∘∥_{2}) is a Banach ∗ - algebra without unit. Analogously, if A ∈ N (H) and {e_{n}} is an orthonormal basis of H then ∑ ⟨Ae_{n}, e_{n}⟩ = ∑ ρ_{n}, i.e. the sum of the former series does not depend on the choice of {e_{n}} . This value is known as the trace of A and it is denote as tr (A) . Further, if we let ∥A∥_{1} = tr (|A|) then (N (H) , ∥∘∥_{1}) is a Banach algebra. If A ∈ B (H) then A ∈ N (H) if and only if |A|^{1/2} ∈ S (H). For more details on this matter the reader can see [7], Ch. I. §2. In this article, by a derivation on a Banach algebra 𝔘 we will mean any linear operator δ : 𝔘 → 𝔘 so that the usual Leibnitz rule δ (a · b) = δ (a) · b + a · δ (b) holds for all a, b ∈ 𝔘. In particular, given a ∈ 𝔘 the maps [a, b] = a · b − b · a defined for all b ∈ 𝔘 are the so called inner derivations. We will say that a derivation is outer if it is not inner.

The authors previously studied the existence and structure of general derivations in the frame of weight sequence Banach algebras (cf. [1]). Our matter in this article is to consider questions concerning to innerness of bounded derivations on S (H). This problem has been solved in other contexts; for instance in the frame of von Neumann algebras every bounded derivation is inner (cf. [10], [11]). In Section 2 we consider the structure of general (bounded) derivations on S(H). We develop, in Th. 2 and Prop. 5, the intrinsic relationship between bounded derivations on S(H) and derivations on a Hilbert space of infinite complex matrices (cf. [2], [3]). The precise structure of derivations on S(H) is given in Prop. 7 and Prop. 8 allows us to define an equivalence relation on them. So, in Corollary 11 we see how each infinite matrix derivation determines a unique equivalence class of bounded derivations on S(H). In particular, from this development it follows the simple structure of the so called Hadamard derivations. Finally, in Section 3 we describe the action of bounded derivations on self-adjoint Hilbert-Schmidt operators. Hadamard derivations and their restrictions to nuclear operators on H are considered in Prop. 15. In Prop. 16 we realize certain derivations on S(H), in general not inner, as certain series of inner derivations on S(H).

ON INNERNESS OF DERIVATIONS ON S(H) 23

Notation 1. Throughout this article ω will be an infinite countable set and, if 𝔸 is a Banach algebra, D(𝔸) will denote the class of bounded derivations on 𝔸. Let l^{2} (ω × ω) be the Hilbert space of infinite matrices

a = (a_{n,m})_{n,m∈ω} endowed with the norm ∥a∥_{2} = (∑_{n,m∈ω} |a_{n,m}|^{2})^{1/2}. We will write by means of \overline{a}, a^{t} and a^{∗} the conjugate, the transpose and the adjoint of a respectively. If m, n ∈ ω it is easy to see that a_{n,m}^{∗} = \overline{a_{m,n}}, as usual a_{n,m}^{t} = a_{m,n} and \overline{a} = a^{∗t}.

2. On the structure of general derivations on S(H)

Theorem 2. (cf. [2], [4]) A bounded linear endomorphism Δ of l^{2} (ω × ω) is a derivation if and only if there are matrices α = {α_{n,m}}_{n,m∈ω} and β = {β_{n,m}}_{n,m∈ω} of complex numbers uniquely determined so that

(i) For any n ∈ ω, α_{n,n} = 0.

(ii) sup_{n,m∈ω} |β_{n,m}| < ∞.

(iii) The matrix α is nearly-inner, i.e. the formal mapping L_{α} : z →

α · z − z · α defines a bounded linear operator on l^{2} (ω × ω) . (iv) For any n, m, p ∈ ℕ the identities β_{n,m} + β_{m,p} = β_{n,p} hold. Then

Δ (z) = ∑_{n,m∈ω} (∑_{p∈ω}(z_{p,m} α_{n,p} − α_{p,m} z_{n,p}) + z_{n,m} β_{n,m}) e_{n,m} (1)

for z ∈ l^{2} (ω × ω), where for all n, m ∈ ω we are writing e_{n,m} = (δ_{n,m}^{p,q})_{p,q∈ω} and δ_{n,m}^{p,q} denotes the usual Kronecker´ s function. In particular, we can denote Δ = Δ_{α,β}.

Remark 3. Let Δ ∈ D (l^{2} (ω × ω)). So, the corresponding entries of the matrices α and β related to Δ according to Th.2 are defined by the relations

Δ (e_{n,m}) = ∑_{p∈ω}(α_{p,n} e_{p,m} − α_{m,p} e_{n,p}) + β_{n,m} e_{n,m}, n, m ∈ ω.

In particular, observe that

sup_{n,m∈ω} ∑_{p∈ω} (|α_{p,n}|^{2} + |α_{m,p}|^{2}) + |β_{n,m}|^{2} ≤ ∥Δ∥^{2} < ∞,

i.e. all rows and columns of a nearly-inner matrix are bounded and square summable.

Proposition 4. Let α, α_{1}, α_{2} be nearly-inner matrices with null diagonals and let β, β_{1}, β_{2} be bounded matrices that verify condition (iv) of Th.2. So, the following formulae hold:

(i) [Δ_{α_{1},0}, Δ_{α_{2},0}] = Δ_{α(α_{1},α_{2}),β(α_{1},α_{2})}, where for each n, m ∈ ω

\begin{aligned}α (α_{1}, α_{2})_{n,m} & = (1 − δ_{n,m}) · [α_{1}, α_{2}]_{n,m} , \\ β (α_{1}, α_{2})_{n,m} & = [α_{1}, α_{2}]_{n,n} + [α_{2}, α_{1}]_{m,m} . \end{aligned}  (2)

(ii) [Δ_{α,0}, Δ_{0,β}] = −Δ_{α⊙β,0}, where α⊙β = {α_{n,m} · β_{n,m}}_{n,m∈ω} denotes

the Hadamard product of the matrices α and β. (iii) [Δ_{0,β_{1}}, Δ_{0,β_{2}}] = 0. (iv) [Δ_{α_{1},β_{1}}, Δ_{α_{2},β_{2}}] = Δ_{α(α_{1},α_{2})−α_{1}⊙β_{2}+α_{2}⊙β_{1},β(α_{1},α_{2})}.

Proof. Given n, m ∈ ω we obtain that

[Δ_{α_{1},0}, Δ_{α_{2},0}] (e_{n,m})

= ∑_{p∈ω} (δ_{n}^{p} [α_{1}, α_{2}]_{p,n} · e_{p,m} + δ_{m}^{p} [α_{2}, α_{1}]_{m,p} · e_{n,p})

+ ([α_{1}, α_{2}]_{n,n} + [α_{2}, α_{1}]_{m,m}) · e_{n,m}. (3)

By Remark 3 the infinite matrix {[α_{1}, α_{2}]_{n,m}}_{n,m∈ω} is defined. Hence, relations (2) will be established if we show that α (α_{1}, α_{2}) and β (α_{1}, α_{2}) satisfy the conditions of Th.2. For, by Remark 3 and the definitions in (2) they do for β (α_{1}, α_{2}) . By definition α (α_{1}, α_{2}) has null diagonal. In order to see that α (α_{1}, α_{2}) is nearly-inner let z ∈ l^{2} (ω × ω) be a matrix with only a finite number of non zero entries. If n, m ∈ ω we have

(L_{α_{1}} ∘ L_{α_{2}} − L_{α_{2}} ∘ L_{α_{1}}) (z)_{n,m}

= L_{α(α_{1},α_{2})} (z)_{n,m} + z_{n,m} · β (α_{1}, α_{2})_{n,m} . (4)

In consequence, the formal operator L_{α(α_{1},α_{2})} is defined and obviously linear on the dense subspace of matrices with finite support of l^{2} (ω × ω) . Since α_{1} and α_{2} are nearly-inner matrices by (4) the restriction of L_{α(α_{1},α_{2})} to this subspace is continuous. Thus by completeness it extends to a unique bounded operator on l^{2} (ω × ω) , i.e. the matrix α (α_{1}, α_{2}) is nearly-inner and (i) follows. Since L_{α⊙β} = − [L_{α}, Δ_{0,β}] and L_{α}, Δ_{0,β} ∈ B (l^{2} (ω × ω)) then L_{α⊙β} is also bounded. So α ⊙ β is a nearly-inner matrix and as it has null diagonal (ii) holds. Now, the proofs of (iii) and (iv) are straightforward.

Proposition 5. Let e = {e_{n}}_{n∈ω}, f = {f_{m}}_{m∈ω} be orthonormal basis of H, let U ∈ B (H) be the unitary operator so that Ue_{n} = f_{n} if n ∈ ω and let A ∈ S (H) .

(i) If 𝔍_{e,f} (A) = {⟨Ae_{n}, f_{m}⟩}_{n,m∈ω} then 𝔍_{e,f} defines an isometric iso-

morphism of S(H) onto l^{2} (ω × ω) .

(ii) 𝔍_{e,f}(A) = 𝔍_{e,e} (U^{∗}A) = 𝔍_{f,f}(AU^{∗}).

(iii) The map 𝔖_{e,f}(A) = \overline{𝔍_{e,f}(UA^{∗})} is an ∗-algebraic isometric isomor-

phism of S (H) onto l^{2} (ω × ω). Proof. Since

∑_{n,m∈ω} |⟨Ae_{n}, f_{m}⟩|^{2} = ∑_{n∈ω} ∥Ae_{n}∥^{2} = ∥A∥_{2}^{2} < ∞

then 𝔍_{e,f} is well defined and it is clearly an ∗- isometry from S(H) into l^{2} (ω × ω) . If a = (a_{n,m})_{n,m∈ω} belongs to l^{2} (ω × ω) it is easy to see that

h → Ah = ∑_{m} f_{m} ∑_{n} a_{n,m} ⟨h, e_{n}⟩

defines a Hilbert - Schmidt operator on H so that 𝔍_{e,f} (A) = a. Hence by the open mapping theorem (i) holds. For (ii) it suffices to observe that for all n, m ∈ ω is

⟨Ae_{n}, f_{m}⟩ = ⟨Ae_{n}, Ue_{m}⟩ = ⟨AU^{∗}f_{n}, f_{m}⟩ .

Moreover,

𝔖_{e,f}(A)_{n,m}^{∗} = \overline{𝔖_{e,f} (A)_{m,n}} = ⟨UA^{∗}e_{m}, f_{n}⟩
= ⟨e_{m}, Ae_{n}⟩ = \overline{⟨UAe_{n}, f_{m}⟩} = \overline{𝔍_{e,f}(UA)_{n,m}}
= 𝔖_{e,f}(A^{∗})_{n,m}

It is clear that 𝔖_{e,f} is linear and as ∥UA^{∗}∥_{2} = ∥A∥_{2} if A ∈ S (H) by (i) 𝔖_{e,f} becomes an isometry. On the other hand, if a ∈ l^{2} (ω × ω) it is easily seeing that 𝔖_{e,f}^{−1}(a) = 𝔍_{e,f}^{−1} (\overline{a})^{∗} ∘U and 𝔖_{e,f}^{−1} becomes also isometric. Finally, if A, B ∈ S(H) and m, n ∈ ω we have

(𝔖_{e,f}(A) · 𝔖_{e,f}(B))_{n,m} = ∑_{p∈ω} 𝔖_{e,f}(A)_{n,p} · 𝔖_{e,f}(B)_{p,m}

= ∑_{p∈ω} ⟨f_{p}, UA^{∗}e_{n}⟩ · ⟨f_{m}, UB^{∗}e_{p}⟩ = ∑_{p∈ω} ⟨Ae_{p}, e_{n}⟩ · ⟨Be_{m}, e_{p}⟩

= ⟨∑_{p∈ω} ⟨Be_{m}, e_{p}⟩ Ae_{p}, e_{n}⟩ = ⟨ABe_{m}, e_{n}⟩ = ⟨U^{∗}f_{m}, (AB)^{∗}e_{n}⟩

= \overline{⟨U(AB)^{∗}e_{n}, f_{m}⟩} = 𝔖_{e,f} (AB)_{n,m}

and (iii) follows.

Remark 6. In what follows, if a, b ∈ H we will write a^{∗} ⊗ b to denote the vector map

(a^{∗} ⊗ b) (h) = ⟨h, a⟩ · b, (h ∈ H).

It is easy to see that

(1) a^{∗} ⊗ b is a Hilbert-Schmidt if a, b ∈ H since it is a finite rank operator.

(2) ⟨Aa, b⟩ = ⟨A, a^{∗} ⊗ b⟩_{2} if a, b ∈ H and A ∈ S(H).

(3) (a_{1} + a_{2})^{∗} ⊗ b = a_{1}^{∗} ⊗ b + a_{2}^{∗} ⊗ b if a_{1}, a_{2}, b ∈ H .

(4) a^{∗} ⊗ (b_{1} + b_{2}) = a^{∗} ⊗ b_{1} + a^{∗} ⊗ b_{2} if a, b_{1}, b_{2} ∈ H.

(5) (λa)^{∗} ⊗ b = λ̄ (a^{∗} ⊗ b) = a^{∗} ⊗ (λ̄b) if a, b ∈ H and λ ∈ ℂ.

(6) (a^{∗} ⊗ b)^{∗} = b^{∗} ⊗ a if a, b ∈ H.

(7) If e = {e_{n}}_{n∈ω} and f = {f_{m}}_{m∈ω} are orthonormal basis of H the set {e_{n}^{∗} ⊗ f_{m}}_{n,m∈ω} becomes orthonormal basis of S(H). In fact, the class of finite rank operators is dense in S (H), (cf. [7], p. 36).

Proposition 7. Let δ ∈ D (S (H)), A ∈ S (H) and let e = {e_{n}}_{n∈ω} be an orthonormal basis of H . There exist a unique set of bounded linear forms {γ_{e}^{n,m}}_{n,m∈ω} on S (H) so that δ (A) can be written in S (H) as

δ (A) = ∑_{m,n∈ω}γ_{e}^{n,m} (A) e_{n}^{∗} ⊗ e_{m}.  (5)

Further, there exist unique matrices α and β as in Th. 2 so that for each n, m ∈ ω is γ_{e}^{n,m}(A) = ⟨A, B_{e}^{n,m}⟩_{2},

B_{e}^{n,m} = e_{n}^{∗} ⊗ [e(ᾱ_{m}) + β̄_{m,n} · e_{m}] − e(α^{n})^{∗} ⊗ e_{m},

e(\overline{α}_{m}) = ∑_{p∈ω} \overline{α_{m,p}} · e_{p} and α^{n} = ∑_{p∈ω} α_{p,n} · e_{p}.

Proof. Given two orthonormal basis e = {e_{n}}_{n∈ω}, f = {f_{m}}_{m∈ω} of H, by (iii) of Prop. 5 there is a 1-1 correspondence

Ψ_{e,f} : D (l^{2} (ω × ω)) → D(S(H)), Ψ_{e,f} (Δ) = 𝔖_{e,f}^{−1} ∘ Δ ∘ 𝔖_{e,f}. (6)

So, if δ ∈ D(S(H)) there are unique matrices α = (α_{m,n})_{m,n∈ω} , β = (β_{m,n})_{m,n∈ω} in the conditions of Th. 2 so that 𝔖_{e,f} ∘ δ ∘ 𝔖_{e,f}^{−1} = Δ_{α,β}. Now, with the above notation if A ∈ S (H) then

δ (A) = (𝔖_{e,f}^{−1} ∘ Δ) (\overline{𝔍_{e,f}(UA^{∗})})

= ∑_{m,n∈ω} (∑_{p∈ω}(⟨Ae_{n}, e_{p}⟩ α_{m,p} − α_{p,n} ⟨Ae_{p}, e_{m}⟩) + ⟨Ae_{n}, e_{m}⟩ β_{m,n})

· 𝔖_{e,f}^{−1} (e_{m,n}) (7)

= ∑_{m,n∈ω} (⟨Ae_{n}, e (\overline{α}_{m})⟩ − ⟨Ae (α^{n}) , e_{m}⟩ + ⟨Ae_{n}, e_{m}⟩ β_{m,n}) · e_{n}^{∗} ⊗ e_{m}

= ∑_{m,n∈ω} ⟨A, B_{e}^{n,m}⟩_{2} · e_{n}^{∗} ⊗ e_{m}

and so (5) follows.

Proposition 8. Let e = {e_{n}}_{n∈ω} , f = {f_{m}}_{m∈ω} , g = {g_{n}}_{n∈ω} , h = {h_{m}}_{m∈ω} be orthonormal basis of H, Δ ∈ D (l^{2} (ω × ω)), A ∈ S(H).Then

Ψ_{e,f} (Δ) (V^{∗}AU) = V^{∗}Ψ_{g,h} (Δ) (A) U,  (8)

where U, V ∈ B(H) are the unitary operators so that Ue_{n} = g_{n} and V f_{m} = h_{m} if n, m ∈ ω.

Proof. Observe that both sides in (8) are defined because S(H) is a two-sided ideal of B(H) (cf. [6], §15.4.8, p. 335). The proof follows observing that

e_{n}^{∗} ⊗ f_{m} = V^{∗} (g_{n}^{∗} ⊗ h_{m}) U and γ_{e,f}^{n,m} (V^{∗}AU) = γ_{g,h}^{n,m}(A)

for each m, n ∈ ω and all A ∈ S(H).

Notation 9. Let δ_{1}, δ_{2} ∈ D (S (H)) . We will write δ_{1} ∼ δ_{2} if and only if there are unitary operators U, V on H so that δ_{1} (V^{∗}AU) = V^{∗}δ_{2} (A) U if A ∈ S (H) . It is readily seeing that ∼ defines an equivalence relation on D (S (H)) .

Corollary 10. Given δ_{1}, δ_{2} ∈ D(S(H)), δ_{1} ∼ δ_{2} if and only if for all orthonormal basis e = {e_{n}}_{n∈ω} , f = {f_{m}}_{m∈ω} of H there are orthonormal basis g = {g_{n}}_{n∈ω} , h = {h_{m}}_{m∈ω} of H so that Ψ_{e,f}^{−1} (δ_{1}) = Ψ_{g,h}^{−1}(δ_{2}).

Proof.

(⇒): Let U, V unitary operators on H so that δ_{1} (V^{∗}AU) = V^{∗}δ_{2} (A) U for all A ∈ S (H) . Given orthonormal basis e = {e_{n}}_{n∈ω} , f = {f_{m}}_{m∈ω} of H there exists Δ ∈ D (l^{2} (ω × ω)) so that Ψ_{e,f} (Δ) = δ_{1}. If we write Ue_{n} = g_{n} and V f_{m} = h_{m} if n, m ∈ ω then g = {g_{n}}_{n∈ω} , h = {h_{m}}_{m∈ω} are orthonormal basis of H. So, by Prop. 8 we have Ψ_{g,h} (Δ) = δ_{2} and the condition is necessary. (⇐): Let e = {e_{n}}_{n∈ω} , g = {g_{n}}_{n∈ω} , h = {h_{m}}_{m∈ω} be fixed orthonormal basis of H so that Ψ_{e,e}^{−1} (δ_{1}) and Ψ_{g,h}^{−1}(δ_{2}) determine a same element Δ in D (l^{2} (ω × ω)) . Hence δ_{1} ∼ δ_{2} since by Prop. 8 there are unitary operators U, V ∈ B(H) so that for all A ∈ S(H) is

δ_{1} (V^{∗}AU) = Ψ_{e,e} (Δ) (V^{∗}AU) = V^{∗}Ψ_{g,h} (Δ) (A) U = V^{∗}Ψ_{g,h} (Δ) (A) U.

Corollary 11. There is a 1-1 correspondence between D(S(H))/ ∼ onto D (l^{2} (ω × ω)) , i.e. D(S(H))/ ∼ ≡ D (l^{2} (ω × ω)) .

Proof. Let us write

Ψ : D (l^{2} (ω × ω)) → D(S(H))/ ∼, Ψ (Δ) = [Ψ_{e,f} (Δ)]^{∼} ,

where Δ ∈ D (l^{2} (ω × ω)) , e and f are orthonormal basis of H and [Ψ_{e,f} (Δ)]^{∼} denotes the ∼ equivalence class of Ψ_{e,f} (Δ) in D(S(H))/ ∼ . By Prop. 8 the function Ψ is well defined, i.e. Ψ (Δ) does not depend on the choice of the orthonormal basis e nor f. If Δ_{1}, Δ_{2} ∈ D (l^{2} (ω × ω)) and Ψ (Δ_{1}) = Ψ (Δ_{2}) then Ψ_{e,e} (Δ_{1}) = Ψ_{e,e} (Δ_{2}) . By Prop. 8 we have Δ_{1} = Δ_{2}, i.e.Ψ is injective. We have already seen that for any orthonormal basis e and f of H the map Ψ_{e,f}, introduced in (6), defines a bijection between D (l^{2} (ω × ω)) and D(S(H)). Therefore Ψ is also surjective.

Notation 12. From now on we’ll denote any bounded derivation δ on D(S(H)) as δ = δ_{α,β}, where α, β are the unique infinite matrices determined by δ as we have pointed out in Prop. 7. This matrices are uniquely by means of Th. 2 and they identify the corresponding ∼ equivalence class of δ as in Corollary 11. So, δ is intrinsically determined by these matrices while its coordinate representation changes according to the rules of Prop. 4 and (8) of Prop. 8. In particular, we shall say that any bounded derivation δ_{0,β} on S (H) is a Hadamard derivation.

Remark 13. Any Hadamard derivation is induced by bounded sequences of complex numbers. For, if {b_{n}}_{n∈ω} is such a sequence and

b = {b_{n} − b_{m}}_{n,m∈ω} ,

then δ_{o,b} is a Hadamard derivation. On the other hand, given δ_{0,β} we already know that β_{n,m} = β_{n,l} + β_{l,m} for all n, l, m ∈ ω. In consequence β has null diagonal and β_{n,m} = β_{n,1} + β_{1,m} = −β_{1,n} + β_{1,m}, i.e. the first row determines the whole β. Of course, although β defines δ_{0,β} uniquely it does not give rise to a unique bounded sequence.

3. On certain particular derivations

Proposition 14. Let δ_{α,β} ∈ D (S (H)) and A be a self-adjoint Hilbert-Schmidt operator on H. If e = {e_{n}}_{n∈ω} is the orthonormal basis of H induced by the sequence {ρ_{n}}_{n∈ω} of eigenvalues of A then

δ_{α,β} (A) = ∑_{n∈ω} e_{n}^{∗} ⊗ (A − ρ_{n} · Id_{H}) e (α_{n})

= − ∑_{n∈ω} ((A − ρ_{n} · Id_{H}) e (\overline{α}^{n}))^{∗} ⊗ e_{n}

Proof. If n ∈ ω then

δ_{α,β} (e_{n}^{∗} ⊗ e_{n}) = Ψ_{e,e} (Δ_{α,β}) (e_{n}^{∗} ⊗ e_{n})  (9)

= 𝔖_{e,e}^{−1} (Δ_{α,β} (e_{n,n})) = ∑_{m∈ω} (α_{m,n} e_{n}^{∗} ⊗ e_{m} − α_{n,m} e_{m}^{∗} ⊗ e_{n})

Since A = ∑_{n∈ω} ρ_{n} e_{n}^{∗} ⊗ e_{n} in S (H) by (9) we obtain that

δ_{α,β} (A) = ∑_{n,m∈ω} α_{n,m} · (ρ_{m} − ρ_{n}) e_{n}^{∗} ⊗ e_{m}.  (10)

Now from (10) our claim follows easily.

Proposition 15. (i) Any Hadamard derivation on S (H) is the re-

striction of an inner one on B (H) . (ii) The restriction of any Hadamard derivation to N (H) belongs to

D(N (H)). Proof.

(i) Let δ_{0,β} be a Hadamard derivation and let e = {e_{n}}_{n∈ω} be an orthonormal basis in H. By Remark (13) we can assume that β = {β_{m} − β_{n}}_{m,n∈ω}, where {β_{n}}_{n∈ω} is a bounded sequence in ℂ. Let us prove that if A ∈ S(H) then

δ_{0,β} (A) = [∑_{p∈ω} (\overline{β}_{p} · e_{p})^{∗} ⊗ e_{p}, A] .  (11)

For, if n, m ∈ ω then

[∑_{p∈ω} (\overline{β}_{p} · e_{p})^{∗} ⊗ e_{p}, e_{n}^{∗} ⊗ e_{m}] = (β_{m} − β_{n}) · (e_{n}^{∗} ⊗ e_{m})

= ∑_{p,q∈ω} γ_{e}^{p,q} (e_{n}^{∗} ⊗ e_{m}) e_{p}^{∗} ⊗ e_{q}

= δ_{0,β} (e_{n}^{∗} ⊗ e_{m}) ,

i.e. (11) holds by Prop.7. Indeed, δ_{0,β} (∘) = [𝔥 (β) , ∘] , where 𝔥 (β) ∈ B(H) is the diagonal operator 𝔥 (β) = ∑_{p∈ω} β_{p}·(e_{p}^{∗} ⊗ e_{p}) .

(ii) With the notation of (i), since N (H) is an ideal our claim is clear if 𝔥 (β) ∈ N (H) (for instance, if {β_{n}}_{n∈ω} ∈ l^{1} (ω)). For the general case, let A ∈ N (H). Actually, let A = U ∘ |A| be the polar decomposition of A and {ρ_{n}}_{n∈ω} the sequence of eigenvalues of |A| associated to an orthonormal basis e = {e_{n}}_{n∈ω} . Since

|A| = ∑_{n∈ω} ρ_{n} · (e_{n}^{∗} ⊗ e_{n}) in S(H), we get

δ_{0,β}(|A|) = ∑_{n∈ω} ρ_{n} · δ_{0,β} (e_{n}^{∗} ⊗ e_{n}) = 0

as follows by Prop. 14. Consequently we see that

δ_{0,β}(A) = [𝔥 (β) , U ∘ |A|] = [𝔥 (β) , U] ∘ |A| + U ∘ [𝔥 (β) , |A|] = [𝔥 (β) , U] ∘ |A| . (12)

But for all S, T ∈ B(H) we have ∥SAT ∥_{1} ≤ ∥S∥ ∥A∥_{1} ∥T ∥ (cf. [9], §3.34 (xi), p. 133). Accordingly we obtain

∥δ_{0,β}(A)∥_{1} ≤ ∥[𝔥 (β) , U]∥ · ∥A∥_{1} ≤ 2 ∥𝔥 (β)∥ · ∥A∥_{1}

and ∥𝔥 (β)∥ = sup_{n∈ω} |β_{n}| < ∞, i.e. δ_{0,β} |_{N (H)} ∈ D(N (H)).

Proposition 16. Let α be a nearly-inner matrix with null diagonal. Given an orthonormal basis e = {e_{n}}_{n∈ω} of H and A ∈ S(H) then

δ_{α,0} (A) = ∑_{p∈ω} [e_{p}^{∗} ⊗ e (α^{p}) , A] .  (13)

Proof. By Prop. 7, (7) we have

δ_{α,0} (A) = ∑_{n,m∈ω} ∑_{p∈ω} (⟨Ae_{n}, e_{p}⟩ α_{m,p} − α_{p,n} ⟨Ae_{p}, e_{m}⟩) · e_{n}^{∗} ⊗ e_{m}. (14)

Hence, if A belongs to the finite linear span of {e_{n}^{∗} ⊗ e_{m}}_{n,m∈ω} by (14) we obtain that

δ_{α,0} (A) = ∑_{p∈ω} {(A^{∗}e_{p})^{∗} ⊗ e (α^{p}) − e_{p}^{∗} ⊗ Ae (α^{p})} .  (15)

Since {e_{n}^{∗} ⊗ e_{m}}_{n,m∈ω} is a basis of the complete space S(H) and δ_{α,0} ∈ B(S(H)) then (15) holds for all A ∈ S(H). Now, it is straightforward to see that each summand in (15) defines a bounded derivation on S(H). Indeed, those derivations are all inner, if p ∈ ω is

(A^{∗}e_{p})^{∗} ⊗ e (α^{p}) − e_{p}^{∗} ⊗ Ae (α^{p}) = [e_{p}^{∗} ⊗ e (α^{p}) , A]

and we get (13).

Example 17. Let H = l^{2} (ℕ) and let α be the nearly-inner matrix so that α_{n,m} is 0 or 1, according as m ≠ n + 1 or m = n + 1 respectively. On identifying e_{0}^{∗} with the zero form on H then α induces the bounded derivation

δ_{α,0} (A) = ∑_{n≥1,m≥1} ⟨A, e_{n}^{∗} ⊗ e_{m+1} − e_{n−1}^{∗} ⊗ e_{m}⟩_{2} · e_{n}^{∗} ⊗ e_{m}

defined for A ∈ S(H). Let us assume that δ_{α,0} is inner, say δ_{α,0} (∘) = [C, ∘] for same C ∈ S(H). So, if r, s ∈ ℕ we get

[C, e_{r}^{∗} ⊗ e_{s}] = e_{r}^{∗} ⊗ Ce_{s} − (C^{∗}e_{r})^{∗} ⊗ e_{s}  (16)

and

δ_{α,0} (e_{r}^{∗} ⊗ e_{s}) = \{ \begin{matrix}−e_{r+1}^{∗} ⊗ e_{1} & if & s = 1, \\ e_{r}^{∗} ⊗ e_{s−1} − e_{r+1}^{∗} ⊗ e_{s} & if & s > 1. \end{matrix}  (17)

From (16) and (17), if r ≥ 1 and s > 1 we deduce that

[C, e_{r}^{∗} ⊗ e_{s}] (e_{r}) = Ce_{s} − ⟨Ce_{r}, e_{r}⟩ · e_{s} = e_{s−1}.

Hence if r ≠ s is ⟨Ce_{s}, e_{r}⟩ = δ_{r}^{s−1}. Thus {⟨Ce_{s}, e_{r}⟩}_{s,r∈ℕ} becomes not square summable, which contradicts Prop. 5. So δ_{α,0} is not inner, although it is realized as a generalized series of inner derivations as stated in Prop. 16.

References

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[2] Barrenechea, A. L. & Peña, C. C.: Examples of bounded derivations on some none algebras. Actas del VII Congreso Dr. A. Monteiro, 23 – 25, (2004).

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32 A. L. BARRENECHEA, C. C. PEÑA

Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA02111-1307, USA, (http://www.smm.org.mx/SMMP).

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UNCentro – FCExactas – NuCoMPA – Dpto. de Matemáticas. Tandil, Pcia, de Buenos Aires, Argentina E-mail address: ccpenia@exa.unicen.edu.ar E-mail address: analucia@exa.unicen.edu.ar

Received June 1, 2005; revised version August 8, 2005