On the Approximation Numbers

of Large Toeplitz Matrices

A. Böttcher^{∗}

Received: January 14, 1997

Communicated by Alfred K. Louis

Abstract. The kth approximation number s_{k}^{(p)}(A_{n}) of a complex n × n matrix A_{n} is defined as the distance of A_{n} to the n × n matrices of rank at most n − k. The distance is measured in the matrix norm associated with the l^{p} norm (1 < p < ∞) on C^{n}. In the case p = 2, the approximation numbers coincide with the singular values.

We establish several properties of s_{k}^{(p)}(A_{n}) provided A_{n} is the n × n truncation of an infinite Toeplitz matrix A and n is large. As n → ∞, the behavior of s_{k}^{(p)}(A_{n}) depends heavily on the Fredholm properties (and, in particular, on the index) of A on l^{p}.

This paper is also an introduction to the topic. It contains a concise history of the problem and alternative proofs of the theorem by G. Heinig and F. Hellinger as well as of the scalar-valued version of some recent results by S. Roch and B. Silbermann concerning block Toeplitz matrices on l^{2}.

1991 Mathematics Subject Classification: Primary 47B35; Secondary 15A09, 15A18, 15A60, 47A75, 47A58, 47N50, 65F35

1. Introduction

Throughout this paper we tacitly identify a complex n × n matrix with the operator it induces on C^{n}. For 1 < p < ∞, we denote by C_{p}^{n} the space C^{n} with the l^{p} norm,

∥x∥_{p} := (|x_{1}|^{p} + … + |x_{n}|^{p})^{1/p},

and given a complex n × n matrix A_{n}, we put

∥A_{n}∥_{p} := sup_{x≠0} (∥A_{n}x∥_{p}/∥x∥_{p}).  (1)

∗Research supported by the Alfried Krupp Förderpreis für junge Hochschullehrer of the Krupp Foundation

Documenta Mathematica 2 (1997) 1–29

We let B(C_{p}^{n}) stand for the Banach algebra of all complex n × n matrices with the norm (1). For j ∈ {0, 1, …, n}, let F_{j}^{(n)} be the collection of all complex n × n matrices of rank at most j, i.e., let

F_{j}^{(n)} := {F ∈ B(C_{p}^{n}) : dim Im F ≤ j}.

The kth approximation number (k ∈ {0, 1, …, n}) of A_{n} ∈ B(C_{p}^{n}) is defined as

s_{k}^{(p)}(A_{n}) := dist (A_{n}, F_{n−k}^{(n)}) := min {∥A_{n} − F_{n}∥_{p} : F_{n} ∈ F_{n−k}^{(n)}}.  (2)

(note that F_{j}^{(n)} is a closed subset of B(C_{p}^{n})). Clearly,

0 = s_{0}^{(p)}(A_{n}) ≤ s_{1}^{(p)}(A_{n}) ≤ … ≤ s_{n}^{(p)}(A_{n}) = ∥A_{n}∥_{p}.

It is easy to show (see Proposition 9.2) that

s_{1}^{(p)}(A_{n}) = \{ \begin{matrix}1/∥A_{n}^{−1}∥_{p} & if & A_{n} & is invertible, \\ 0 & if & A_{n} & is not invertible. \end{matrix}  (3)

Notice also that in the case p = 2 the approximation numbers s_{1}^{(2)}(A_{n}), …, s_{n}^{(2)}(A_{n}) are just the singular values of A_{n}, i.e., the eigenvalues of (A_{n}^{∗}A_{n})^{1/2}.

Let T be the complex unit circle and let a ∈ L^{∞} := L^{∞}(T). The n × n Toeplitz matrix T_{n}(a) generated by a is the matrix

T_{n}(a) := (a_{j−k})_{j,k=1}^{n}  (4)

where a_{l} (l ∈ Z) is the lth Fourier coefficient of a,

a_{l} := \frac{1}{2π} ∫_{0}^{2π}a(e^{iθ})e^{−ilθ}dθ.

This paper is devoted to the limiting behavior of the numbers s_{k}^{(p)}(T_{n}(a)) as n goes to infinity.

Of course, the study of properties of T_{n}(a) as n → ∞ leads to the consideration of the infinite Toeplitz matrix

T (a) := (a_{j−k})_{j,k=1}^{∞}.

The latter matrix induces a bounded operator on l^{2} := l^{2}(N) if (and only if) a ∈ L^{∞}. Acting with T (a) on l^{p} := l^{p}(N) is connected with a multiplier problem in case p ≠ 2. We let M_{p} stand for the set of all a ∈ L^{∞} for which T (a) generates a bounded operator on l^{p}. The norm of this operator is denoted by ∥T (a)∥_{p}. The function a is usually referred to as the symbol of T (a) and T_{n}(a).

In this paper, we prove the following results.

Theorem 1.1. If a ∈ M_{p} then for each k,

s_{n−k}^{(p)}(T_{n}(a)) → ∥T (a)∥_{p} as n → ∞.

Theorem 1.2. If a ∈ M_{p} and T (a) is not normally solvable on l^{p} then for each k,

s_{k}^{(p)}(T_{n}(a)) → 0 as n → ∞

Let M_{⟨2⟩} := L^{∞}. For p ≠ 2, we define M_{⟨p⟩} as the set of all functions a ∈ L^{∞} which belong to Mp̃ for all p̃ in some open neighborhood of p (which may depend on a). A well known result by Stechkin says that a ∈ M_{p} for all p ∈ (1, ∞) whenever a ∈ L^{∞} and the total variation V_{1}(a) of a is finite and that in this case

∥T (a)∥_{p} ≤ C_{p}(∥a∥_{∞} + V_{1}(a))  (5)

with some constant C_{p} < ∞ (see, e.g., [5, Section 2.5(f)] for a proof). We denote by P C the closed subalgebra of L^{∞} constituted by all piecewise continuous functions. Thus, a ∈ P C if and only if a ∈ L^{∞} and the one-sided limits

a(t ± 0) := lim_{ε→0±0} a(e^{i(θ+ε)})

exist for every t = e^{iθ} ∈ T. By virtue of (5), the intersection P C ∩ M_{⟨p⟩} contains all piecewise continuous functions of finite total variation.

Throughout what follows we define q ∈ (1, ∞) by 1/p + 1/q = 1 and we put

[p, q] := [ min{p, q}, max{p, q}].

One can show that if a ∈ M_{p}, then a ∈ M_{r} for all r ∈ [p, q] (see, e.g., [5, Section 2.5(c)]).

Here is the main result of this paper.

Theorem 1.3. Let a be a function in P C ∩ M_{⟨p⟩} and suppose T (a) is Fredholm of the same index −k (∈ Z) on l^{r} for all r ∈ [p, q]. Then

lim_{n→∞} s_{|k|}^{(p)}(T_{n}(a)) = 0 and lim inf_{n→∞} s_{|k|+1}^{(p)}(T_{n}(a)) > 0.

For p = 2, Theorems 1.2 and 1.3 are special cases of results by Roch and Silbermann [20], [21]. Since a Toeplitz operator on l^{2} with a piecewise continuous symbol is either Fredholm (of some index) or not normally solvable, Theorems 1.2 and 1.3 completely identify the approximation numbers (= singular values) which go to zero in the case p = 2.

Now suppose p ≠ 2. If a ∈ C ∩ M_{⟨p⟩}, then T (a) is again either Fredholm or not normally solvable, and hence Theorems 1.2 and 1.3 are all we need to see which approximation numbers converge to zero. In the case where a ∈ P C ∩ M_{⟨p⟩} we have three mutually excluding possibilities (see Section 3):

(i) T (a) is Fredholm of the same index −k on l^{r} for all r ∈ [p, q];

(ii) T (a) is not normally solvable on l^{p} or not normally solvable on l^{q}; (iii) T (a) is normally solvable on l^{p} and l^{q} but not normally solvable on l^{r} for some

r ∈ (p, q) := [p, q] \ {p, q}.

In the case (i) we can apply Theorem 1.3. Since

s_{k}^{(p)}(T_{n}(a)) = s_{k}^{(q)}(T_{n}(a))  (6)

(see (35)), Theorem 1.2 disposes of the case (ii). I have not been able to settle the case (iii). My conjecture is as follows.

Conjecture 1.4. In the case (iii) we have

s_{k}^{(p)}(T_{n}(a)) → 0 as n → ∞

for every fixed k.

The paper is organized as follows. Section 2 is an attempt at presenting a short history of the topic. In Section 3 we assemble some results on Toeplitz operators on l^{p} which are needed to prove the three theorems stated above. Their proofs are given in Sections 4 to 6. The intention of Sections 7 and 8 is to illustrate how some simple constructions show a very easy way to understand the nature of the Heinig/Hellinger and Roch/Silbermann results. Notice, however, that the approach of Sections 7 and 8 cannot replace the methods of these authors. They developed some sort of high technology which enabled them to tackle the block case and more general approximation methods, while in these two sections it is merely demonstrated that in the scalar case (almost) all problems can be solved with the help of a few crowbars (Theorems 7.1, 7.2, 7.4). Nevertheless, beginners will perhaps appreciate reading Sections 7 and 8 before turning to the papers [13] and [25], [20].

2. Brief history

The history of the lowest approximation number s_{1}^{(p)}(T_{n}(a)) is the history of the finite section method for Toeplitz operators: by virtue of (3), we have

s_{1}^{(p)}(T_{n}(a)) → 0 ⟺ ∥T_{n}^{−1}(a)∥_{p} → ∞.

We denote by Φ_{k}(l^{p}) the collection of all Fredholm operators of index k on l^{p}. The equivalence

lim sup_{n→∞} ∥T_{n}^{−1}(a)∥_{p} < ∞ ⟺ T (a) ∈ Φ_{0}(l^{p})  (7)

was proved by Gohberg and Feldman [7] in two cases: if a ∈ C ∩M_{⟨p⟩} (where C stands for the continuous functions on T) or if p = 2 and a ∈ P C. For a ∈ P C ∩ M_{⟨p⟩}, the equivalence

lim sup_{n→∞} ∥T_{n}^{−1}(a)∥_{p} < ∞ ⟺ T (a) ∈ Φ_{0}(l^{r}) for all r ∈ [p, q]  (8)

holds. This was shown by Verbitsky and Krupnik [30] in the case where a has a single jump, by Silbermann and the author [3] for symbols with finitely many jumps, and finally by Silbermann [23] for symbols with a countable number of jumps. In the work of many authors, including Ambartsumyan, Devinatz, Shinbrot, Widom, Silbermann, it was pointed out that (7) is also true if

p = 2 and a ∈ (C + H^{∞}) ∪ (C + \overline{H^{∞}}) ∪ P QC

(see [4], [5]). Also notice that the implication “⟹” of (8) is valid for every a ∈ M_{p}. Treil [26] proved that there exist symbols a ∈ M_{⟨2⟩} = L^{∞} such that T (a) ∈ Φ_{0}(l^{2}) but ∥T_{n}^{−1}(a)∥_{2} is not uniformly bounded; concrete symbols with this property can be found in the recent article [2, Section 7.7].

The Toeplitz matrices

T_{n}(φ_{γ}) = (\frac{1}{j − k + γ} )_{j,k=1}^{n} (γ ∉ Z)

are the elementary building blocks of general Toeplitz matrices with piecewise continuous symbols and have therefore been studied for some decades. The symbol is given by

φ_{γ}(e^{iθ}) = \frac{π}{sin πγ} e^{iπγ}e^{−iγθ}, θ ∈ [0, 2π).

This is a function in P C with a single jump at e^{iθ} = 1. Tyrtyshnikov [27] focussed attention on the singular values of T_{n}(φ_{γ}). He showed that

s_{1}^{(2)}(T_{n}(φ_{γ})) = O(1/n^{|γ|−1/2}) if γ ∈ R and |γ| > 1/2

and that there are constants c_{1}, c_{2} ∈ (0, ∞) such that

c_{1}/ log n ≤ s_{1}^{(2)}(T_{n}(φ_{1/2})) ≤ c_{2}/ log n.

Curiously, the case |γ| < 1/2 was left as an open problem in [27], although from the standard theory of Toeplitz operators with piecewise continuous symbols it is well known that

T (φ_{γ}) ∈ Φ_{0}(l^{2}) ⟺ |Re γ| < 1/2

(see, e.g., [7, Theorem IV.2.1] or [5, Proposition 6.24]), which together with (7) (for p = 2 and a ∈ P C) implies that

lim inf_{n→∞} s_{1}^{(2)}(T_{n}(φ_{γ})) = 0 if |Re γ| ≥ 1/2  (9)

and

lim inf_{n→∞} s_{1}^{(2)}(T_{n}(φ_{γ})) > 0 if |Re γ| < 1/2

(see [20]). A simple and well known argument (see the end of Section 3) shows that in (9) the liminf can actually be replaced by lim.

Also notice that it was already in the seventies when Verbitsky and Krupnik [30] proved that

lim_{n→∞} s_{1}^{(p)}(T_{n}(φ_{γ})) = 0 ⟺ |Re γ| ≥ min{1/p, 1/q}

(full proofs are also in [4, Proposition 3.11] and [5, Theorem 7.37; in part (iii) of that theorem there is a misprint: the −1/p < Re β < 1/q must be replaced by −1/q < Re β < 1/p]).

As far as I know, collective phenomena of s_{1}^{(p)}(T_{n}(a)), …, s_{n}^{(p)}(T_{n}(a)) have been studied only for p = 2, and throughout the rest of this section we abbreviate s_{k}^{(2)}(T_{n}(a)) to s_{k}(T_{n}(a)).

In 1920, Szegö showed that if a ∈ L^{∞} is real-valued and F is continuous on R, then

\frac{1}{n} ∑_{k=1}^{n} F (s_{k}(T_{n}(a))) → \frac{1}{2π} ∫_{0}^{2π}F (|a(e^{iθ})|) dθ.  (10)

In the eighties, Parter [15] and Avram [1] extended this result to arbitrary (complexvalued) symbols a ∈ L^{∞}. Formula (10) implies that

{s_{k}(T_{n}(a))}_{k=1}^{n} and {|a(e^{2πik/n})|}_{k=1}^{n}  (11)

are equally distributed (see [9] and [29]).

Research into the asymptotic distribution of the singular values of Toeplitz matrices was strongly motivated by a phenomenon discovered by C. Moler in the middle of the eighties. Moler observed that almost all singular values of T_{n}(φ_{1/2}) are concentrated in [π − ε, π] where ε is very small. Formula (10) provides a way to understand this phenomenon: letting F = 1 on [0, π − 2ε] and F = 0 on [π − ε, π] and taking into account that |φ_{1/2}| = 1, one gets

\frac{1}{n} ∑_{k=1}^{n} F (s_{k}(T_{n}(φ_{1/2}))) → \frac{1}{2π} ∫_{0}^{2π}F (1) dθ = F (1) = 0,

which shows that the percentage of the singular values of T_{n}(φ_{1/2}) which are located in [0, π − 2ε] goes to zero as n increases to infinity.

Widom [32] was the first to establish a second order result on the asymptotics of singular values. Under the assumption that

a ∈ L^{∞} and ∑_{n∈Z} |n| |a_{n}|^{2} < ∞

and that F ∈ C^{3}(R), he showed that

∑_{k=1}^{n} F (s_{k}^{2}(T_{n}(a))) = \frac{n}{2π} ∫_{0}^{2π}F (|a(e^{iθ})|^{2}) dθ + E_{F} (a) + o(1)

with some constant E_{F} (a), and he gave an expression for E_{F} (a). He also introduced two limiting sets of the sets

Σ(T_{n}(a)) := {s_{1}(T_{n}(a)), …, s_{n}(T_{n}(a))}, which, following the terminology of [19], are defined by

Λ_{part} (Σ(T_{n}(a))) := {λ ∈ R : λ is partial limit of some sequence
{λ_{n}} with λ_{n} ∈ Σ(T_{n}(a))},
Λ_{unif} (Σ(T_{n}(a))) := {λ ∈ R : λ is the limit of some sequence
{λ_{n}} with λ_{n} ∈ Σ(T_{n}(a))}.

It turned out that for large classes of symbols a we have

Λ_{part} (Σ(T_{n}(a))) = Λ_{unif} (Σ(T_{n}(a))) = sp (T (\overline{a})T (a))^{1/2}  (12)

where sp A := {λ ∈ C : A − λI is not invertible} denotes the spectrum of A (on l^{2}) and \overline{a} is defined by \overline{a}(e^{iθ}) := \overline{a(e^{iθ})}. Note that T (\overline{a}) is nothing but the adjoint T^{∗}(a) of T (a). Widom [32] proved (12) under the hypothesis that a ∈ P C or that a is locally self-adjoint, while Silbermann [24] derived (12) for locally normal symbols. Notice that symbols in P C or even in P QC are locally normal.

In the nineties, Tyrtyshnikov [28], [29] succeeded in proving that the sets (11) are equally distributed under the sole assumption that a ∈ L^{2} := L^{2}(T). His approach is based on the observation that if ∥A_{n} − B_{n}∥_{F} = o(n), where ∥ · ∥_{F} stands for the Frobenius (or Hilbert-Schmidt) norm, then A_{n} and B_{n} have equally distributed singular values. The result mentioned can be shown by taking A_{n} = T_{n}(a) and choosing appropriate circulants for B_{n}.

The development received a new impetus from Heinig and Hellinger’s 1994 paper [13]. They considered normally solvable Toeplitz operators on l^{2} and studied the problem whether the Moore-Penrose inverses of T_{n}^{+}(a) of T_{n}(a) converge strongly on l^{2} to the Moore-Penrose inverse T^{+}(a) of T (a). Recall that the Moore-Penrose inverse of a normally solvable Hilbert space operator A is the (uniquely determined) operator A^{+} satisfying

AA^{+}A = A, A^{+}AA^{+} = A^{+}, (A^{+}A)^{∗} = A^{+}A, (AA^{+})^{∗} = AA^{+}.

If a ∈ C, then T (a) is normally solvable on l^{2} if and only if a(t) ≠ 0 for all t ∈ T. When writing T_{n}^{+}(a) → T^{+}(a), we actually mean that T_{n}^{+}(a)P_{n} → T^{+}(a), where P_{n} is the projection defined by

P_{n} : {x_{1}, x_{2}, x_{3}, …} ↦ {x_{1}, x_{2}, …, x_{n}, 0, 0, …}.  (13)

It is not difficult to verify that T_{n}^{+}(a) → T^{+}(a) strongly on l^{2} if and only if T (a) is normally solvable and

lim sup_{n→∞} ∥T_{n}^{+}(a)∥_{2} < ∞.  (14)

Heinig and Hellinger investigated normally solvable Toeplitz operators T (a) with symbols in the Wiener algebra W,

a ∈ W ⟺ ∥a∥_{W} := ∑_{n∈Z} |a_{n}| < ∞,

and they showed that then (14) is satisfied if and only if there is an n_{0} ≥ 1 such that

Ker T (a) ⊂ Im P_{n_{0}} and Ker T (\overline{a}) ⊂ Im P_{n_{0}},  (15)

where Ker A := {x ∈ l^{2} : Ax = 0} and Im A := {Ax : x ∈ l^{2}}. (This formulation of the Heinig-Hellinger result is due to Silbermann [25].) Conditions (15) are obviously met if T (a) is invertible, in which case even ∥T_{n}^{−1}(a)∥_{2} is uniformly bounded. The really interesting case is the one in which T (a) is not invertible, and in that case (15) and thus (14) are highly instable. For example, if a is a rational function (without poles on T) and λ ∈ sp T (a), then

lim sup_{n→∞} ∥T_{n}^{+}(a − λ)∥_{2} < ∞  (16)

can only hold if λ belongs to sp T_{n}(a) for all sufficiently large n. Consequently, (16) implies that λ lies in Λ_{unif}(sp T_{n}(a)), and the latter set is extremely “thin”: it is contained in a finite union of analytic arcs (see [22] and [6]).

What has Moore-Penrose invertibility to do with singular values ? The answer is as follows: if A_{n} ∈ B(C_{2}^{n}) and s_{k}(A_{n}) is the smallest nonzero singular value of A_{n}, then

∥A_{n}^{+}∥_{2} = 1/s_{k}(A_{n}).

Thus, (14) holds exactly if there exists a d > 0 such that

Σ(T_{n}(a)) ⊂ {0} ∪ [d, ∞)  (17)

for all sufficiently large n.

Now Silbermann enters the scene. He replaced the Heinig-Hellinger problem by another one. Namely, given T (a), is there a sequence {B_{n}} of operators B_{n} ∈ B(C_{2}^{n}) with the following properties: there exists a bounded operator B on l^{2} such that

B_{n} → B and B_{n}^{∗} → B^{∗} strongly on l^{2}

and

∥T_{n}(a)B_{n}T_{n}(a) − T_{n}(a)∥_{2} → 0, ∥B_{n}T_{n}(a)B_{n} − B_{n}∥_{2} → 0,
∥(B_{n}T_{n}(a))^{∗} − B_{n}T_{n}(a)∥_{2} → 0, ∥(T_{n}(a)B_{n})^{∗} − T_{n}(a)B_{n}∥_{2} → 0 ?

Such a sequence {B_{n}} is referred to as an asymptotic Moore-Penrose inverse of T (a). In view of the (instable) conditions (15), the following result by Silbermann [25] is surprising: if a ∈ P C and T (a) is normally solvable, then T (a) always has an asymptotic Moore-Penrose inverse. And what is the concern of this result with singular values ? One can easily show T (a) has an asymptotic Moore-Penrose inverse if and only if there is a sequence c_{n} → 0 and a number d > 0 such that

Σ(T_{n}(a)) ⊂ [0, c_{n}] ∪ [d, ∞).  (18)

One says that Σ(T_{n}(a)) has the splitting property if (18) holds with c_{n} → 0 and d > 0. Thus, Silbermann’s result implies that if a ∈ P C and T (a) is normally solvable on l^{2}, then Σ(T_{n}(a)) has the splitting property.

Only recently, Roch and Silbermann [20], [21] were able to prove even much more. The sets Σ(T_{n}(a)) are said to have the k-splitting property, where k ≥ 0 is an integer, if (18) is true for some sequence c_{n} → 0 and some d > 0 and, in addition, exactly k singular values lie in [0, c_{n}] and n − k singular values are located in [d, ∞)

(here multiplicities are taken into account). Equivalently, Σ(T_{n}(a)) has the k-splitting property if and only if

lim_{n→∞} s_{k}(T_{n}(a)) = 0 and lim inf_{n→∞} s_{k+1}(T_{n}(a)) > 0.  (19)

A normally solvable Toeplitz operator T (a) on l^{2} with a symbol a ∈ P C is automatically Fredholm and therefore has some index k ∈ Z. Roch and Silbermann [20], [21] discovered that then Σ(T_{n}(a)) has the |k|-splitting property. In other words, if a ∈ P C and T (a) ∈ Φ_{k}(l^{2}) then (19) holds with k replaced by |k|. Notice that this Theorem 1.3 for p = 2.

In fact, it was the Roch and Silbermann papers [20], [21] which stimulated me to do some thinking about singular values. It was the feeling that the |k|-splitting property must have its root in the possibility of “ignoring |k| dimensions” which led me to the observation that none of the works cited in this section makes use of the fact that s_{k}(A_{n}) may alternatively be defined by (2), i.e. that singular values may also be viewed as approximation numbers. I then realized that some basic phenomena of [20] and [21] can be very easily understood by having recourse to (2) and that, moreover, using (2) is a good way to pass from l^{2} and C^{∗}-algebras to l^{p} and Banach algebras.

3. Toeplitz operators on l^{p}

We henceforth always assume that 1 < p < ∞ and 1/p + 1/q = 1.

Let M_{p} and M_{⟨p⟩} be as in Section 1. The set M_{p} can be shown to be a Banach algebra with pointwise algebraic operations and the norm ∥a∥_{M_{p}} := ∥T (a)∥_{p}. It is also well known that

M_{p} = M_{q} ⊂ M_{2} = L^{∞}

and

∥a∥_{M_{p}} = ∥a∥_{M_{q}} ≥ ∥a∥_{M_{2}} = ∥a∥_{∞}  (20)

(see, e.g., [5, Section 2.5]). We remark that working with M_{⟨p⟩} instead of M_{p} is caused by the need of somehow reversing the estimate in (20). Suppose, for instance, p > 2 and a ∈ M_{⟨p⟩}. Then a ∈ M_{p+ε} for some ε > 0, and the Riesz-Thorin interpolation theorem gives

∥a∥_{M_{p}} ≤ ∥a∥_{M_{2}}^{γ} ∥a∥_{p+ε}^{1−γ} = ∥a∥_{∞}^{γ} ∥a∥_{M_{p+ε}}^{1−γ}  (21)

with some γ ∈ (0, 1) depending only on p and ε. The ∥a∥_{M_{p+ε}} on the right of (21) may in turn be estimated by C_{p}(∥a∥_{∞} + V_{1}(a)) (recall Stechkin’s inequality (5)) provided a has bounded total variation.

A bounded linear operator A on l^{p} is said to be normally solvable if its range, Im A, is a closed subset of l^{p}. The operator A is called Fredholm if it is normally solvable and the spaces

Ker A := {x ∈ l^{p} : Ax = 0} and Coker A := l^{p}/Im A

have finite dimensions. In that case the index Ind A is defined as

Ind A := dim Ker A − dim Coker A.

We denote by Φ(l^{p}) the collection of all Fredholm operators on l^{p} and by Φ_{k}(l^{p}) the operators in Φ(l^{p}) whose index is k. The following four theorems are well known. Comments are at the end of this section.

Theorem 3.1. Let a ∈ M_{p}.

(a) If a does not vanish identically, then the kernel of T (a) on l^{p} or the kernel of T (\overline{a}) on l^{q} is trivial.

(b) The operator T (a) is invertible on l^{p} if and only if T (a) ∈ Φ_{0}(l^{p}).

Of course, part (b) is a simple consequence of part (a).

Theorem 3.2. Let a ∈ C ∩ M_{⟨p⟩}. Then T (a) is normally solvable on l^{p} if and only if a(t) ≠ 0 for all t ∈ T. In that case T (a) ∈ Φ(l^{p}) and

Ind T (a) = −wind a,

where wind a is the winding number of a about the origin.

Now let a ∈ P C, t ∈ T, and suppose a(t − 0) ≠ a(t + 0). We denote by

A_{p}(a(t − 0), a(t + 0))

the circular arc at the points of which the line segment [a(t−0), a(t+0)] is seen at the angle max{2π/p, 2π/q} and which lies on the right of the straight line passing first a(t − 0) and then a(t + 0) if 1 < p < 2 and on the left of this line if 2 < p < ∞. For p = 2, A_{p}(a(t−0), a(t+0)) is nothing but the line segment [a(t−0), a(t+0)] itself. Let a_{p}^{#} denote the closed, continuous, and naturally oriented curve which results from the (essential) range R(a) of a by filling in the arcs A_{p}(a(t − 0), a(t + 0)) for each jump. In case this curve does not pass through the origin, we let wind a_{p}^{#} be its winding number.

Theorem 3.3. Let a ∈ P C ∩ M_{⟨p⟩}. Then T (a) is normally solvable on l^{p} if and only if 0 ∉ a_{p}^{#}. In that case T (a) ∈ Φ(l^{p}) and

Ind T (a) = −wind a_{p}^{#}.

For a ∈ P C and t ∈ T, put
O_{p}(a(t − 0), a(t + 0)) := ⋃_{r∈[p,q]} A_{r}(a(t − 0), a(t + 0)).  (22)

If a(t − 0) ≠ a(t + 0) and p ≠ 2, then O_{p}(a(t − 0), a(t + 0)) is a certain lentiform set. Also for a ∈ P C, let

a_{[p,q]}^{#} := ⋃_{r∈[p,q]} a_{r}^{#}.

Thus, a_{[p,q]}^{#} results from R(a) by filling in the sets (22) between the endpoints of the

jumps. If 0 ∉ a_{[p,q]}^{#}, then necessarily 0 ∉ a_{2}^{#} and we define wind a_{[p,q]}^{#} as wind a_{2}^{#} in this case.

From Theorem 3.3 we deduce that the conditions (i) to (iii) of Section 1 are equivalent to the following:

(i’) 0 ∉ a_{[p,q]}^{#} and wind a_{[p,q]}^{#} = k;

(ii’) 0 ∈ a_{p}^{#} ∪ a_{q}^{#};

(iii’) 0 ∈ a_{[p,q]}^{#} \ (a_{p}^{#} ∪ a_{q}^{#}).

For a ∈ M_{p}, let T_{n}(a) ∈ B(C_{p}^{n}) be the operator given by the matrix (4). One says that the sequence {T_{n}(a)} := {T_{n}(a)}_{n=1}^{∞} is stable if

lim sup_{n→∞} ∥T_{n}^{−1}(a)∥_{p} < ∞.

Here we follow the practice of putting

∥T_{n}^{−1}(a)∥_{p} = ∞ if T_{n}(a) is not invertible.

In other words, {T_{n}(a)} is stable if and only if T_{n}(a) is invertible for all n ≥ n_{0} and there exists a constant M < ∞ such that ∥T_{n}^{−1}(a)∥_{p} ≤ M for all n ≥ n_{0}. From (3) we infer that

{T_{n}(a)} is stable ⟺ lim inf_{n→∞} s_{1}^{(p)}(T_{n}(a)) > 0.

Theorem 3.4. (a) If a ∈ C ∩ M_{⟨p⟩} then

{T_{n}(a)} is stable ⟺ 0 ∉ a(T) and wind a = 0.

(b) If a ∈ P C ∩ M_{⟨p⟩} then

{T_{n}(a)} is stable ⟺ 0 ∉ a_{[p,q]}^{#} and wind a_{[p,q]}^{#} = 0.

As already said, these theorems are well known. Theorem 3.1 is due to Coburn (p = 2) and Duduchava (p ≠ 2), Theorem 3.2 is Gohberg and Feldman’s, Theorem 3.3 is the result of many authors in the case p = 2 and was established by Duduchava for p ≠ 2, Theorem 3.4 goes back to Gohberg and Feldman for a ∈ C ∩ M_{⟨p⟩} (general p) and a ∈ P C (p = 2), and it was obtained in the work of Verbitsky, Krupnik, Silbermann, and the author for a ∈ P C ∩ M_{⟨p⟩} and p ≠ 2. Precise historical remarks and full proofs are in [5].

Part (a) of Theorem 3.4 is clearly a special case of part (b). In fact, Theorem 3.4(b) may also be stated as follows: {T_{n}(a)} contains a stable subsequence {T_{n_{j}}(a)} (n_{j} → ∞) if and only if 0 ∉ a_{[p,q]}^{#} and wind a_{[p,q]}^{#} = 0. Hence, we arrive at the conclusion that if a ∈ P C ∩ M_{⟨p⟩}, then

s_{1}^{(p)}(T_{n}(a)) → 0
⟺ {T_{n}(a)} is stable
⟺ 0 ∈ a_{[p,q]}^{#} or (0 ∉ a_{[p,q]}^{#} and wind a_{[p,q]}^{#} ≠ 0).

At this point the question of whether the lowest approximation number of T_{n}(a) goes to zero or not is completely disposed of for symbols a ∈ P C ∩ M_{⟨p⟩}.

4. Proof of Theorem 1.1.

Contrary to what we want, let us assume that there is a c < ∥T (a)∥_{p} such that s_{n−k}^{(p)}(T_{n}(a)) ≤ c for all n in some infinite set N. Since s_{n−k}^{(p)}(T_{n}(a)) = dist (T_{n}(a), F_{k}^{(n)}), we can find F_{n} ∈ F_{k}^{(n)} (n ∈ N) so that ∥T_{n}(a) − F_{n}∥_{p} ≤ c. For x = (x_{1}, …, x_{n}) and y = (y_{1}, …, y_{n}), we define

(x, y) := x_{1}y_{1} + … + x_{n}y_{n}.  (23)

By [16, Lemma B.4.11], there exist e_{j}^{(n)} ∈ C_{p}^{n}, f_{j}^{(n)} ∈ C_{p}^{n}, γ_{j}^{(n)} ∈ C such that

F_{n}x = ∑_{j=1}^{k} γ_{j}^{(n)}(x, f_{j}^{(n)})e_{j}^{(n)} (x ∈ C_{p}^{n}),

∥e_{j}^{(n)}∥_{p} = 1, ∥f_{j}^{(n)}∥_{q} = 1, and

|γ_{j}^{(n)}| ≤ ∥F_{n}∥_{p} ≤ ∥T_{n}(a)∥_{p} + ∥F_{n} − T_{n}(a)∥_{p} ≤ ∥T (a)∥_{p} + c  (24)

for all j ∈ {1, …, k}.

Fix x ∈ C_{p}^{n}, y ∈ C_{q}^{n} and suppose ∥x∥_{p} = 1, ∥y∥_{q} = 1. We then have

|(T_{n}(a)x, y) − ∑_{j=1}^{k} γ_{j}^{(n)}(x, f_{j}^{(n)})(e_{j}^{(n)}, y)| ≤ ∥T_{n}(a) − F_{n}∥_{p} ≤ c.  (25)

Clearly, (T_{n}(a)x, y) → (T (a)x, y). From (24) and the Bolzano-Weierstrass theorem we infer that the sequence {(γ_{1}^{(n)}, …, γ_{k}^{(n)})}_{n∈N} has a converging subsequence. Without loss of generality suppose the sequence itself converges, i.e.

(γ_{1}^{(n)}, …, γ_{k}^{(n)}) → (γ_{1}, …, γ_{k}) ∈ C^{k}

as n ∈ N goes to infinity. The vectors e_{j}^{(n)} and f_{j}^{(n)} all belong to the unit sphere of l^{p} and l^{q}, respectively. Hence, by the Banach-Alaoglu theorem (see, e.g., [18, Theorem IV.21]), {e_{j}^{(n)}}_{n∈N} and {f_{j}^{(n)}}_{n∈N} have subsequences converging in the weak ∗-topology. Again we may without loss of generality assume that

e_{j}^{(n)} → e_{j} ∈ l^{p}, f_{j}^{(n)} → f_{j} ∈ l^{q}

in the weak ∗-topology as n ∈ N goes to infinity.

From (25) we now obtain that if x ∈ l^{p} and y ∈ l^{q} have finite support and ∥x∥_{p} = 1, ∥y∥_{q} = 1, then

|(T (a)x, y)) − ∑_{j=1}^{k} γ_{j}(x, f_{j})(e_{j}, y)| ≤ c.

This implies that

∥T (a) − F ∥_{p} ≤ c  (26)

where F is the finite-rank operator given by

F x := ∑_{j=1}^{k} γ_{j}(x, f_{j})e_{j} (x ∈ l^{p}).  (27)

Let ∥T (a)∥^{(ess)} denote the essential norm of T (a) on l^{p}, i.e. the distance of T (a) to the compact operators on l^{p}. By (26) and (27),

∥T (a)∥_{p}^{(ess)} ≤ ∥T (a) − F ∥_{p} ≤ c < ∥T (a)∥_{p}.

However, one always has ∥T (a)∥_{p}^{(ess)} = ∥T (a)∥_{p} (see, e.g., [5, Proposition 4.4(d)]). This contradiction completes the proof.

5. Proof of Theorem 1.2.

We will employ the following two results.

Theorem 5.1. Let A be a bounded linear operator on l^{p}.

(a) The operator A is normally solvable on l^{p} if and only if

k_{A} := sup_{x∈l^{p}, ∥x∥_{p}=1} dist (x, Ker A) < ∞.

(b) If M is a closed subspace of l^{p} and dim (l^{p}/M) < ∞, then the normal solvability of A|M : M → l^{p} is equivalent to the normal solvability of A : l^{p} → l^{p}.

A proof is in [8, pp. 159–160].

Theorem 5.2. If M is a k-dimensional subspace of C_{p}^{n}, then there exists a projection Π : C_{p}^{n} → C_{p}^{n} such that Im Π = M and ∥Π∥_{p} ≤ k.

This is a special case of [16, Lemma B.4.9].

Theorem 1.2 is trivial in case a vanishes identically. So suppose a ∈ M_{p} \{0} and T (a) is not normally solvable on l^{p}. Then the adjoint operator T (\overline{a}) is not normally solvable on l^{q}. By Theorem 3.1(a), Ker T (a) = \{0\} on l^{p} or Ker T (\overline{a}) = \{0\} on l^{q}.

Since s_{k}^{(p)}(T_{n}(a)) = s_{k}^{(q)}(T_{n}(\overline{a})), we may a priori assume that Ker T (a) = \{0\} on l^{p}. Abbreviate T (a) and T_{n}(a) to A and A_{n}, respectively.

Define P_{n} on l^{p} by (13) and let

V := l^{p} → l^{p}, {x_{1}, x_{2}, x_{3}, …} ↦ {0, x_{1}, x_{2}, x_{3}, …}.

As A|Im V^{n} : Im V^{n} → l^{p} has the same matrix as AV^{n} : l^{p} → l^{p}, we deduce from Theorem 5.1(b) that there is no n ≥ 0 such that AV^{n} is normally solvable. Note that Ker (AV^{n}) = {0} for all n ≥ 0.

Let l^{p}(n_{1}, n_{2}] denote the sequences {x_{j}}_{j=1}^{∞} ∈ l^{p} which are supported in (n_{1}, n_{2}], i.e., for which x_{j} = 0 whenever j ≤ n_{1} or j > n_{2}.

Lemma 5.3. There are 0 = N_{0} < N_{1} < N_{2} < … and z_{j} ∈ l^{p}(N_{j−1}, N_{j}] (j ≥ 1) such that

∥z_{j}∥_{p} = 1 and ∥Az_{j}∥_{p} → 0 as j → ∞.

Proof. By Theorem 5.1(a), there is a y_{1} ∈ l^{p} such that ∥y_{1}∥_{p} = 2 and ∥Ay_{1}∥ < 1/2. If N_{1} is large enough, then ∥P_{N_{1}}y_{1}∥_{p} ≥ 1 and ∥AP_{N_{1}}y_{1}∥_{p} < 1. Letting z_{1} := P_{N_{1}y_{1}}/∥P_{N_{1}}y_{1}∥_{p} we get

z_{1} ∈ l^{p}(0, N_{1}], ∥z_{1}∥_{p} = 1, ∥Az_{1}∥_{p} < 1.

Applying Theorem 5.1(a) to the operator AV^{N_{1}}, we see that there is an y_{2} ∈ l^{p} such that ∥y_{2}∥_{p} = 2 and ∥AV^{N_{1}}y_{2}∥_{p} < 1/4. For sufficiently large N_{2} > N_{1} we have ∥P_{N_{2}}V^{N_{1}}y_{2}∥_{p} ≥ 1 and ∥AP_{N_{2}}V^{N_{1}}y_{2}∥_{p} < 1/2. Setting

z_{2} := P_{N_{2}}V^{N_{1}}y_{2}/∥P_{N_{2}}V^{N_{1}}y_{2}∥_{p},

we therefore obtain

z_{2} ∈ l^{p}(N_{1}, N_{2}], ∥z_{2}∥_{p} = 1, ∥Az_{2}∥_{p} < 1/2.

Continuing in this way we find z_{j} satisfying

z_{j} ∈ l^{p}(N_{j−1}, N_{j}], ∥z_{j}∥_{p} = 1, ∥Az_{j}∥_{p} < 1/j.

Contrary to the assertion of Theorem 1.2, let us assume that there exist k ≥ 1 and d > 0 such that s_{k}^{(p)}(A_{n}) ≥ d for infinitely many n. We may without loss of generality assume that

s_{k}^{(p)}(A_{n}) ≥ d for all n ≥ n_{0}.  (28)

Let ε > 0 be any number such that

2εk^{2} < d.  (29)

Choose z_{j} as in Lemma 5.3. Obviously, there are sufficiently large j and N such that

∥P_{N}z_{l}∥_{p} ≥ 1/2, ∥AP_{N}z_{l}∥_{p} < ε for l ∈ {j + 1, …, j + k}.  (30)

Since P_{N}z_{l} ∈ l^{p}(N_{l−1, l}], it is clear that P_{N}z_{j+1}, …, P_{N}z_{j+k} are linearly independent. Now let n ≥ N. By Theorem 5.2, there is a projection Π_{n} of C_{p}^{n} onto span{P_{N}z_{j+1}, …, P_{N}z_{j+k}} for which ∥Π_{n}∥_{p} ≤ k. Let I_{n} stand for the identity operator on C_{p}^{n}. The space Im (I_{n} − Π_{n}) = Ker Π_{n} has the dimension n − k and hence, I_{n} − Π_{n} ∈ F_{n−k}^{(n)}. Every x ∈ C_{p}^{n} can be uniquely written in the form

x = γ_{1}P_{N}z_{j+1} + … + γ_{k}P_{N}z_{j+k} + w with w ∈ Ker Π_{n}.

Thus,

∥A_{n}x − A_{n}(I_{n} − Π_{n})x∥_{p} = ∥A_{n}Π_{n}x∥_{p}
= ∥γ_{1}A_{n}(P_{N}z_{j+1}) + … + γ_{k}A_{n}(P_{N}z_{j+k})∥_{p} ≤ |γ_{1}|ε + … + |γ_{k}|ε, (31)

the estimate resulting from (30). Taking into account that the sequences P_{N}z_{l} have pairwise disjoint supports, we obtain from (30) that

∥Π_{n}x∥_{p}^{p} = ∥γ_{1}P_{N}z_{j+1} + … + γ_{k}P_{N}z_{j+k}∥_{p}^{p}
= |γ_{1}|^{p}∥P_{N}z_{j+1}∥_{p}^{p} + … + |γ_{k}|^{p}∥P_{N}z_{j+k}∥_{p}^{p}
≥ (1/2)^{p}(|γ_{1}|^{p} + … + |γ_{k}|^{p}) ≥ (1/2)^{p} max_{1≤m≤k} |γ_{m}|^{p}.  (32)

Combining (31) and (32) we get

∥A_{n}x − A_{n}(I_{n} − Π_{n})x∥_{p} ≤ εk max_{1≤m≤k} |γ_{m}| ≤ 2εk∥Π_{n}x∥_{p} ≤ 2εk^{2}∥x∥_{p},

whence s_{k}^{(p)}(A_{n}) = dist (A_{n}, F_{n−k}^{(n)}) ≤ ∥A_{n} − A_{n}(I − Π_{n})∥_{p} ≤ 2εk^{2}. By virtue of (29), this contradicts (28) and completes the proof.

6. Proof of Theorem 1.3.

The Hankel operator on l^{p} induced by a function a ∈ M_{p} is given by the matrix

H(a) = (a_{j+k−1})_{j,k=1}^{∞}.

For a ∈ M_{p}, define ã ∈ M_{p} by ã(e^{iθ}) := a(e^{−iθ}). Clearly,

H(ã) = (a_{−j−k+1})_{j,k=1}^{∞}.

It is well known and easily seen that

T (ab) = T (a)T (b) + H(a)H(b̃)  (33)

for every a, b ∈ M_{p}. A finite section analogue of formula (33) reads

T_{n}(ab) = T_{n}(a)T_{n}(b) + P_{n}H(a)H(b̃)P_{n} + W_{n}H(ã)H(b)W_{n},  (34)

where P_{n} is as in (13) and W_{n} is defined by

W_{n} : {x_{1}, x_{2}, x_{3}, …} ↦ {x_{n}, x_{n−1}, …, x_{1}, 0, 0, …}.

The identity (34) first appeared in Widom’s paper [31], a proof is also in [4, Proposition 3.6] and [5, Proposition 7.7].

We remark that T_{n}(ã) is the transposed matrix of T_{n}(a) and that the identity T_{n}(ã) = W_{n}T_{n}(a)W_{n} holds. In particular, we have

s_{k}^{(q)}(T_{n}(a)) = min {∥T_{n}(a) − F_{n−k}∥_{q} : F_{n−k} ∈ F_{n−k}^{(n)}}
= min {∥T_{n}(ã) − G_{n−k}∥_{p} : G_{n−k} ∈ F_{n−k}^{(n)}}
= min {∥W_{n}(T_{n}(ã) − G_{n−k})W_{n}∥_{p} : G_{n−k} ∈ F_{n−k}^{(n)}}
= min {∥T_{n}(a) − H_{n−k}∥_{p} : H_{n−k} ∈ F_{n−k}^{(n)}}
= s_{k}^{(p)}(T_{n}(a))  (35)

(note also that W_{n} is an invertible isometry on C_{p}^{n}).

To prove Theorem 1.3, we need the following two (well known) lemmas.

Lemma 6.1. If A, B, C ∈ B(C_{p}^{n}) then

s_{k}^{(p)}(ABC) ≤ ∥A∥_{p} s_{k}^{(p)}(B)∥C∥_{p} for all k.

This follows easily from the definition of s_{k}^{(p)}.

Lemma 6.2. If b ∈ M_{p} and {T_{n}(b)} is stable on l^{p}, then T (b) is invertible on l^{p} and T_{n}^{−1}(b) (:= T_{n}^{−1}(b)P_{n}) converges strongly on l^{p} to T^{−1}(b).

This is obvious from the estimates

∥T_{n}^{−1}(b)P_{n}y − T^{−1}(b)y∥_{p}

≤ ∥T_{n}^{−1}(b)∥_{p} ∥P_{n}y − T_{n}(b)P_{n}T^{−1}(b)y∥_{p} + ∥P_{n}T^{−1}(b)y − T^{−1}(b)y∥_{p},

∥x∥_{p} ≤ lim inf_{n→∞} ∥T_{n}^{−1}(b)∥_{p} ∥T (b)x∥_{p}, ∥ξ∥_{q} ≤ lim inf_{n→∞} ∥T_{n}^{−1}(b̃)∥_{q} ∥T (b̃)ξ∥_{q}.

We now establish two propositions which easily imply Theorem 1.3.

Define χ_{k} by χ_{k}(e^{iθ}) = e^{ikθ}. Using Theorem 3.1(b) and formula (33) one can readily see that if a ∈ M_{p}, then T (a) ∈ Φ_{−k}(l^{p}) if and only if a = bχ_{k} and T (b) is invertible on l^{p}.

Propostion 6.3. If b ∈ M_{p} and {T_{n}(b)} is stable on l^{p} then for every k ∈ Z,

lim inf_{n→∞} s_{|k|+1}^{(p)}(T_{n}(bχ_{k})) > 0.

Proof. We can assume that k ≥ 0, since otherwise we may pass to adjoints. Because ∥T_{n}(χ_{−k})∥_{p} = 1, we obtain from Lemma 6.1 that

s_{k+1}^{(p)}(T_{n}(bχ_{k})) = s_{k+1}^{(p)}(T_{n}(bχ_{k}))∥T_{n}(χ_{−k})∥_{p}
≥ s_{k+1}^{(p)}(T_{n}(bχ_{k})T_{n}(χ_{−k})) = s_{k+1}^{(p)}(T_{n}(b) − P_{n}H(bχ_{k})H(χ_{k})P_{n}),

the latter equality resulting from (34) and the identities H(χ̃_{−k}) = H(χ_{k}) and H(χ_{−k}) = 0. As dim Im H(χ_{k}) = k, we get that F_{k} := P_{n}H(bχ_{k})H(χ_{k})P_{n} ∈ F_{k}^{(n)}, whence

s_{k+1}^{(p)}(T_{n}(b) − F_{k}) = inf {∥T_{n}(b) − F_{k} − G_{n−k−1}∥_{p} : G_{n−k−1} ∈ F_{n−k−1}^{(n)}}
≥ inf {∥T_{n}(b) − H_{n−1}∥_{p} : H_{n−1} ∈ F_{n−1}^{(n)}} = s_{1}^{(p)}(T_{n}(b)).

Since {T_{n}(b)} is stable, we infer from (3) that

lim inf_{n→∞} s_{k+1}^{(p)}(T_{n}(bχ_{k})) ≥ lim inf_{n→∞} s_{1}^{(p)}(T_{n}(b)) > 0.

Proposition 6.4. If b ∈ M_{p} and {T_{n}(b)} is stable on l^{p} then for every k ∈ Z,

lim_{n→∞} s_{|k|}^{(p)}(T_{n}(bχ_{k})) = 0.

Proof. Again we may without loss of generality assume that k ≥ 0. Using (34) and Lemma 6.1 we get

s_{k}^{(p)}(T_{n}(bχ_{k})) = s_{k}^{(p)}(T_{n}(χ_{k})T_{n}(b) + P_{n}H(χ_{k})H(b̃)P_{n})
≤ ∥T_{n}(b)∥_{p} s_{k}^{(p)}(T_{n}(χ_{k}) + P_{n}H(χ_{k})H(b̃)P_{n}T_{n}^{−1}(b)).

Put A_{n} := T_{n}(χ_{k}) + P_{n}H(χ_{k})H(b̃)P_{n}T_{n}^{−1}(b). We have

A_{n} = ( \begin{matrix}∗ & C_{n} \\ I_{n−k} & 0 \end{matrix} ) = ( \begin{matrix}∗ & 0 \\ I_{n−k} & 0 \end{matrix} ) + ( \begin{matrix}0 & C_{n} \\ 0 & 0 \end{matrix} ) =: B_{n} + D_{n},

the blocks being of size k × (n − k), k × k, (n − k) × (n − k), (n − k) × k, respectively. Clearly, B_{n} has rank n − k and thus B_{n} ∈ F_{n−k}^{(n)}. It follows that

s_{k}^{(p)}(A_{n}) = s_{k}^{(p)}(A_{n} − B_{n}) = s_{k}^{(p)}(D_{n}) ≤ ∥D_{n}∥_{p} = ∥C_{n}∥_{p},

and we are left with showing that ∥C_{n}∥_{p} → 0.

Let b_{n} (n ∈ Z) be the Fourier coefficients of b, let e_{j} ∈ l^{p} be the sequence whose only nonzero entry is a unit at the jth position, and recall the notation (23). We have C_{n} = (c_{jl}^{(n)})_{j,l=1}^{k}, and it is easily seen that c_{jl}^{(n)} equals (b_{−k+j−1}, …, b_{−k+j−n}) times the (n − k + l)th column of T_{n}^{−1}(b):

c_{jl}^{(n)} = (b_{−k+j−1} … b_{−k+j−n})T_{n}^{−1}(b)P_{n}e_{n−k+l} = (P_{n}f_{jk}, T_{n}^{−1}(b)P_{n}e_{n−k+l})

where

f_{jk} := {b_{−k+j−1}, b_{−k+j−2}, b_{−k+j−3}, … } = T (χ_{−k+j−1})T (b̃)e_{1} ∈ l^{q}.

Consequently,

c_{jl}^{(n)} = (T_{n}^{−1}(b̃)P_{n}f_{jk}, e_{n−k+l})
= (T^{−1}(b̃)f_{jk}, e_{n−k+l}) + (T_{n}^{−1}(b̃)P_{n}f_{jk} − T^{−1}(b̃)f_{jk}, e_{n−k+l}). (36)

The first term on the right of (36) obviously converges to zero as n → ∞. The second term of (36) is at most

∥T_{n}^{−1}(b̃)P_{n}f_{jk} − T^{−1}(b̃)f_{jk}∥_{q}  (37)

(note that ∥e_{n−k+l}∥_{p} = 1). Our assumptions imply that {T_{n}(b̃)} is stable on l^{q}. We so deduce from Lemma 6.2 that (37) tends to zero as n → ∞.

Thus, each entry of the k ×k matrix C_{n} approaches zero as n → ∞. This implies that ∥C_{n}∥_{p} → 0.

Now let a be as in Theorem 1.3. Since T (a) ∈ Φ_{−k}(l^{r}) for all r ∈ [p, q], we have a = bχ_{k} where T (b) ∈ Φ_{0}(l^{r}) for all r ∈ [p, q]. From Theorems 3.3 and 3.4(b) we conclude that {T_{n}(b)} is stable on l^{p}. The assertions of Theorem 1.3 therefore follows from Propositions 6.3 and 6.4.

We remark that Propositions 6.3 and 6.4 actually yield more than Theorem 1.3. Namely, let Π_{p}^{0} denote the collection of all symbols b ∈ M_{p} for which {T_{n}(b)} is stable on l^{p} and let Π_{p} be the set of all symbols a ∈ M_{p} such that aχ_{−k} ∈ Π_{p}^{0} for some k ∈ Z. Notice that

Π_{p} = Π_{q} ⊂ ⋃_{r∈[p,q]} Π_{r}

and

G(C + H^{∞}) ∪ G(C + \overline{H^{∞}}) ∪ G(P QC) ⊂ Π_{2} ≠ L^{∞},

where G(B) stands for the invertible elements of a unital Banach algebra B. The following corollary is immediate from Propositions 6.3 and 6.4.

Corollary 6.5. If a ∈ Π_{p} and T (a) ∈ Φ_{k}(l^{p}) then

Σ^{(p)}(T_{n}(a)) := {s_{1}^{(p)}(T_{n}(a)), …, s_{n}^{(p)}(T_{n}(a))}

has the |k|-splitting property.

We also note that the proof of Proposition 6.4 gives estimates for the speed of convergence of s_{|k|}^{(p)}(T_{n}(bχ_{k})) to zero. For example, if ∑_{n∈Z} |n|^{μ}|b_{n}| < ∞ (μ > 0), then the finite section method is applicable to T (b) on the space l^{2,μ} of all sequences x = {x_{n}}_{n=1}^{∞} such that

∥x∥_{2,μ} := ( ∑_{n=1}^{∞} n^{2μ}|x_{n}|^{2})^{1/2} < ∞

whenever T (b) is invertible (see [17, pp. 106–107] or [5, Theorem 7.25]). Since

∥e_{n−k−l}∥_{2,−μ} = (n − k + l)^{−μ} = O(n^{−μ}),

the proof of Proposition 6.4 implies the following result.

Corollary 6.6. If ∑_{n∈Z} |n|^{μ}|a_{n}| < ∞ for some μ > 0 and T (a) ∈ Φ_{k}(l^{p}) then

s_{|k|}^{(p)}(T_{n}(a)) = O(n^{−μ}) as n → ∞.

7. Remarks on the Hilbert space case

Some aspects of the asymptotic behavior of the approximation numbers (= singular values) of matrices in B(C_{2}^{n}) can be very easily understood by having recourse to the following well known fact (the “singular value decomposition”).

Theorem 7.1. If A_{n} ∈ B(C_{2}^{n}) then there exist unitary matrices U_{n}, V_{n} ∈ B(C_{2}^{n}) such that A_{n} = U_{n}S_{n}V_{n} where

S_{n} = diag (s_{1}(A_{n}), …, s_{n}(A_{n})).

Here and throughout this section we abbreviate s_{k}^{(2)}(A_{n}) to s_{k}(A_{n}).

To illustrate the usefulness of Theorem 7.1, we give another proof of Theorem 1.2 for p = 2. We still need the following result.

Theorem 7.2. A bounded linear Hilbert space operator A is normally solvable if and only if there is a d > 0 such that

sp (A^{∗}A) ⊂ {0} ∪ [d, ∞).

For a proof see [10], [11], [20].

Theorem 7.3. Let a ∈ L^{∞} and suppose T (a) is not normally solvable on l^{2}. Then s_{k}(T_{n}(a)) → 0 as n → ∞ for each k ≥ 1.

Proof. Assume there is a k ≥ 1 such that s_{k}(T_{n}(a)) does not converge to zero. Let k_{0} be the smallest k with this property. Then there are n_{1} < n_{2} < … and d > 0 such that

s_{k_{0}}(T_{n_{j}}(a)) ≥ d and s_{k}(T_{n_{j}}(a)) → 0 for k < k_{0}.  (38)

To simplify notation, let us assume that n_{j} = j for all j.

Write T_{n}(a) = U_{n}S_{n}V_{n} as in Theorem 7.1. If λ ∉ {0} ∪ [d^{2}, ∞), then (38) implies that S_{n}^{2} − λI_{n} is invertible for all sufficiently large n, say for n ≥ n_{0}, and that

∥(S_{n}^{2} − λI_{n})^{−1}∥_{2} ≤ M(λ)

with some M(λ) < ∞ independent of n. Because

T_{n}^{∗}(a)T_{n}(a) − λI_{n} = V_{n}^{∗}(S_{n}^{2} − λI_{n})V_{n},

it follows that T_{n}^{∗}(a)T_{n}(a) − λI_{n} is invertible for n ≥ n_{0} and that

∥(T_{n}^{∗}(a)T_{n}(a) − λI_{n})^{−1}∥_{2} ≤ M(λ).

Consequently, for every x ∈ l^{2} we have

∥(T^{∗}(a)T (a) − λI)x∥_{2} ≥ (1/M(λ))∥x∥_{2},

which implies that T^{∗}(a)T (a) − λI is invertible. Thus,

sp (T^{∗}(a)T (a)) ⊂ {0} ∪ [d^{2}, ∞),

and Theorem 7.2 shows that T (a) must be normally solvable.

Things are more transparent by invoking a few (harmless) C^{∗}-algebras. Let S denote the C^{∗}-algebra of all sequences {A_{n}} := {A_{n}}_{n=1}^{∞} of operators A_{n} ∈ B(C_{2}^{n}) such that

∥{A_{n}}∥ := sup_{n≥1} ∥A_{n}∥_{2} < ∞,

and let S_{c} be the C^{∗}-algebra of all {A_{n}} ∈ S for which there exists a bounded linear operator A on l^{2} such that A_{n} → A and A_{n}^{∗} → A^{∗} strongly. Finally, let C stand for the sequences {A_{n}} ∈ S for which ∥A_{n}∥_{2} → 0. Clearly, C is a closed two-sided ideal in both S and S_{c}.

Obviously, a sequence {A_{n}} ∈ S is stable if and only if {A_{n}} + C is invertible in S/C. Following [25] and [20], we call a sequence {A_{n}} ∈ S a Moore-Penrose sequence if there exists a sequence {B_{n}} ∈ S such that

{A_{n}B_{n}A_{n} − A_{n}} ∈ C, {B_{n}A_{n}B_{n} − B_{n}} ∈ C,  (39)
{(B_{n}A_{n})^{∗} − B_{n}A_{n}} ∈ C, {(A_{n}B_{n})^{∗} − A_{n}B_{n}} ∈ C.  (40)

An element a of a unital C^{∗}-algebra A is said to be Moore-Penrose invertible if there is an element a^{+} ∈ A such that

aa^{+}a = a, a^{+}aa^{+} = a^{+}, (a^{+}a)^{∗} = a^{+}a, (aa^{+})^{∗} = aa^{+}.

Thus, {A_{n}} ∈ S is a Moore-Penrose sequence if and only if {A_{n}}+C is Moore-Penrose invertible in S/C.

The following result is again from [10], [11], [20].

Theorem 7.4. Let A be a unital C^{∗}-algebra. An element a ∈ A is Moore-Penrose invertible in A if and only if there is a d > 0 such that sp (a^{∗}a) ⊂ {0} ∪ [d, ∞).

The next theorem is Roch and Silbermann’s [20]. The proof given here is different from theirs.

Theorem 7.5. A sequence {A_{n}} ∈ S is a Moore-Penrose sequence if and only if

Σ(A_{n}) = {s_{1}(A_{n}), …, s_{n}(A_{n})}

has the splitting property.

Proof. Write A_{n} = U_{n}S_{n}V_{n} as in Theorem 7.1. We have

∥A_{n}B_{n}A_{n} − A_{n}∥_{2} → 0
⟺ ∥U_{n}S_{n}V_{n}B_{n}U_{n}S_{n}V_{n} − U_{n}S_{n}V_{n}∥_{2} → 0
⟺ ∥S_{n}(V_{n}B_{n}U_{n})S_{n} − S_{n}∥_{2} → 0,

and since analogous equivalences hold for the remaining three sequences in (39) and (40), we arrive at the conclusion that {A_{n}} is a Moore-Penrose sequence if and only if {S_{n}} + C is Moore-Penrose invertible in S/C. By Theorem 7.4, this is equivalent to the condition

sp_{S/C}({S_{n}^{2}} + C) ⊂ {0} ∪ [d^{2}, ∞) for some d > 0.  (41)

Let D ⊂ S denote the sequences {A_{n}} constituted by diagonal matrices A_{n}. From the elementary theory of C^{∗}-algebras we get

sp_{S/C}({S_{n}^{2}} + C) = sp_{D/(D∩C)}({S_{n}^{2}} + D ∩ C).  (42)

Consider the infinite diagonal matrix

diag (S_{1}^{2}, S_{2}^{2}, …) = diag (ϱ_{1}, ϱ_{2}, ϱ_{3}, …)

(here S_{m} ∈ B(C_{2}^{m}) and ϱ_{m} ∈ C). Obviously, the spectrum on the right of (42) coincides with the set P{ϱ_{m}} of the partial limits of the sequence {ϱ_{m}}. Consequently,

(41) holds if and only if

P{ϱ_{m}} ⊂ {0} ∪ [d^{2}, ∞) for some d > 0,

which is easily seen to be equivalent to the splitting property of Σ(A_{n}).

Also as in [20], we call a sequence {A_{n}} ∈ S an exact Moore-Penrose sequence if {A_{n}^{+}} belongs to S; here A_{n}^{+} ∈ B(C_{2}^{n}) is the Moore-Penrose inverse of A_{n}.

Proposition 7.6. Let {A_{n}} be a sequence in S_{c} and let A be the strong limit of A_{n}. Then the following are equivalent:

(i) A_{n}^{+} is strongly convergent;

(ii) A is normally solvable and A_{n}^{+} → A^{+} strongly;

(iii) A is normally solvable and {A_{n}} is an exact Moore-Penrose sequence.

The simple proof is omitted.

The following theorem was by means of different methods established in [20].

Theorem 7.7. A sequence {A_{n}} ∈ S is an exact Moore-Penrose sequence if and only if there is a d > 0 such that

Σ(A_{n}) ⊂ {0} ∪ [d, ∞) for all n ≥ 1.  (43)

Proof. As in the proof of Theorem 7.5 we see that {A_{n}} is an exact Moore-Penrose sequence if and only if {S_{n}} enjoys this property. Write

diag (S_{1}, S_{2}, …) = diag (μ_{1}, μ_{2}, μ_{3}, …)

(where again S_{m} ∈ B(C_{2}^{m}) and μ_{n} ∈ C) and define f : [0, ∞) → [0, ∞) by

f(x) := \{ \begin{matrix}x^{−1} & if & x > 0 \\ 0 & if & x = 0. \end{matrix}

Since

diag (S_{1}^{+}, S_{2}^{+}, …) = diag (f(μ_{1}), f(μ_{2}), f(μ_{3}), …),

we conclude that {S_{n}^{+}} ∈ S if and only if {f(μ_{m})} is a bounded sequence, which is equivalent to (43).

Now let A_{n} = T_{n}(a) with a ∈ L^{∞}. If {T_{n}(a)} is a Moore-Penrose sequence, then T (a) must obviously be normally solvable. Thus, from Theorem 3.3 (for p = 2) and Theorem 1.3 (for p = 2) we deduce that if a ∈ P C, then {T_{n}(a)} is a Moore-Penrose sequence if and only if T (a) is Fredholm.

The following result, which is also taken from [20], characterizes the exact Moore-Penrose sequences constituted by the truncations of an infinite Toeplitz matrix. Our proof is again different from the one of [20].

Theorem 7.8. Let a ∈ P C. Then {T_{n}(a)} is an exact Moore-Penrose sequence if and only if T (a) is Fredholm and

dim Ker T_{n}(a) = |Ind T (a)|  (44)

for all sufficiently large n.

Proof. If {T_{n}(a)} is an exact Moore-Penrose sequence, then T (a) is normally solvable and thus Fredholm. Let T (a) ∈ Φ_{k}(l^{2}). Then

s_{|k|}(T_{n}(a)) → 0 and s_{|k|+1}(T_{n}(a)) ≥ d > 0

by virtue of Theorem 1.3 (for p = 2). Since

dist (T_{n}(a), F_{n−|k|−1}^{(n)}) > 0,

we see that

rank T_{n}(a) ≥ n − |k|.  (45)

From Theorem 7.7 we deduce that {T_{n}(a)} is an exact Moore-Penrose sequence if and only if s_{|k|}(T_{n}(a)) = 0 for all n ≥ n_{0}. Because

s_{|k|}(T_{n}(a)) = dist (T_{n}(a), F_{n−|k|}^{(n)}) and F_{n−|k|}^{(n)} is a closed subset of B(C_{2}^{n}), we have s_{|k|}(T_{n}(a)) = 0 if and only if

rank T_{n}(a) ≤ n − |k|.  (46)

Combining (45) and (46) we obtain that {T_{n}(a)} is an exact Moore-Penrose sequence if and only if T (a) ∈ Φ_{k}(l^{2}) for some k ∈ Z and

dim Ker T_{n}(a) = n − rank T_{n}(a) = |k|

for all n ≥ n_{0}.

8. The Heing-Hellinger theorem

Of course, condition (44) is difficult to check. In this section we give a new proof of the Heinig-Hellinger theorem, which provides a criterion (in terms of only the symbol a) for (44) to hold.

If a ∈ P C and T (a) is Fredholm of index zero and thus invertible, then the sequence {T_{n}(a)} is stable (Theorems 3.3 and 3.4 for p = 2). In this case Σ(T_{n}(a)) ⊂ [d, ∞) and dim Ker T_{n}(a) = 0 for all sufficiently large n and hence each of Theorems 7.7 and 7.8 yields that {T_{n}(a)} is an exact Moore-Penrose sequence; however, we have T_{n}^{+}(a) = T_{n}^{−1}(a) for all sufficiently large n and therefore consideration of Moore-Penrose inverses is not at all necessary in this situation.

The really interesting case is the one in which T (a) is Fredholm of nonzero index. The rest of this section is devoted to the proof of the following result.

Theorem 8.1 (Heinig and Hellinger). Let a ∈ P C. Suppose T (a) is Fredholm on l^{2} and Ind T (a) ≠ 0. If Ind T (a) < 0, then the following are equivalent:

(i) dim Ker T_{n}(a) = |Ind T (a)| for all sufficiently large n;

(ii) Ker T (ã) ⊂ Im P_{n_{0}} for some n_{0} ≥ 1;

(iii) the Fourier coefficients (a^{−1})_{−m} are zero for all sufficiently large m.

If Ind T (a) > 0, then the following are equivalent:

(i’) dim Ker T_{n}(a) = Ind T (a) for all sufficiently large n;

(ii’) Ker T (a) ⊂ Im P_{n_{0}} for some n_{0} ≥ 1;

(iii’) (a^{−1})_{m} = 0 for all sufficiently large m.

For the sake of definiteness, let us assume that Ind T (a) = −k < 0. The proofs of the implications (iii) ⇒ (ii) ⇒ (i) are easy.

Proof of the implication (iii) ⇒ (ii). Let x ∈ Ker T (ã). Then, by (33),

T (ã^{−1})T (ã) = I − H(ã^{−1})H(a), which shows that x = H(ã^{−1})H(a)x, and since H(ã^{−1}) has only a finite number of nonzero rows, it follows that x_{m} = 0 for all sufficiently large m.

Proof of the implication (ii) ⇒ (i). If n is large enough then s_{k+1}(T_{n}(ã)) ≥ d > 0 by Theorem 1.3 (or Proposition 6.3), whence rank T_{n}(ã) > n − k + 1 and thus,

dim Ker T_{n}(ã) < k + 1.  (47)

If x ∈ Ker T (ã) ⊂ Im P_{n_{0}} and n ≥ n_{0}, then T_{n}(ã)P_{n}x = P_{n}T (ã)x = 0, which implies that

dim Ker T_{n}(ã) ≥ dim Ker T (ã) = k  (48)

(recall Theorem 3.1(a) for the last equality). Clearly, equality (i) follows from (47) and (48).

The proof of the implication (i) ⇒ (iii) is less trivial and is based on the following deep theorem. Recall that χ_{n} is defined by χ_{n}(t) = t^{n} for t ∈ T.

Theorem 8.2 (Heinig). Let a ∈ L^{∞} and let k > 0 be an integer. Then

dim Ker T_{n}(a) = k for all sufficiently large n

if and only if a or ã is of the form χ_{p+k}(r + h) where h is a function in H^{∞}, r is a rational function in L^{∞}, r has exactly p poles in the open unit disk D (multiplicities taken into account), r has no pole at the origin, and r(0) + h(0) ≠ 0.

A proof is in [12, Satz 6.2 and formula (8.4)]. Also see [14, Theorem 8.6].

Proof of the implication (i) ⇒ (iii). Let χ_{p+k}(r + h) be the representation of a or ã ensured by Theorem 8.2 and put b := χ_{p+k}(r + h). Denote by α_{1}, …, α_{p} and β_{1}, …, β_{q} the poles of r inside and outside T, respectively. For t ∈ T,

r(t) = \frac{u_{+}(t)}{(t − α_{1}) … (t − α_{p})(t − β_{1}) … (t − β_{q})}
= \frac{t^{−p}v_{+}(t)}{(1 − α_{1}/t) … (1 − α_{p}/t)(1 − t/β_{1}) … (1 − t/β_{q})}

with polynomials u_{+}, v_{+} ∈ H^{∞}. Clearly,

s_{+}(t) := (1 − t/β_{1})^{−1} … (1 − t/β_{q})^{−1} ∈ H^{∞}.

Letting

c_{+}(t) := t^{k}v_{+}(t)s_{+}(t) + t^{p+k}(1 − α_{1}/t) … (1 − α_{p}/t)h(t),

we get

b(t) = (1 − α_{1}/t)^{−1} … (1 − α_{p}/t)^{−1}c_{+}(t).

The function c_{+} lies in H^{∞} and has a zero of order at least k at the origin. Obviously, (1 − α_{1}/t)^{−1} … (1 − α_{p}/t)^{−1} is a function which together with its inverse belongs to \overline{H^{∞}}. If c_{+} would have infinitely many zeros in D, then T (c_{+}) and thus T (b) were not Fredholm (see, e.g., [5, Theorem 2.64]). Hence, c_{+} has only a finite number λ ≥ k of zeros in D. It follows that Ind T (c_{+}) = −λ (again see, e.g., [5, Theorem 2.64]) and therefore Ind T (b) = Ind T (c_{+}) = −λ. If b = a, then λ must equal k. Consequently, c_{+}(z) = z^{k}φ_{+}(z) with φ_{+} and φ_{+}^{−1} in H^{∞}. This implies that

a^{−1}(t) = t^{−k}(1 − α_{1}/t) … (1 − α_{p}/t)φ_{+}^{−1}(t)

has only finitely many nonzero Fourier coefficients with negative index. If b would be equal to ã, it would result that Ind T (ã) is negative, which is impossible due to the equality Ind T (ã) = −Ind T (a).

Corollary 8.3. If a ∈ P C \ C then {T_{n}(a)} is an exact Moore-Penrose sequence on l^{2} if and only if {T_{n}(a)} is stable on l^{2}.

Proof. The “if part” is trivial. To prove the “only if” portion, suppose {T_{n}(a)} is an exact Moore-Penrose sequence. Then T (a) is Fredholm by Theorem 7.8. If T (a) has index zero, then {T_{n}(a)} is stable. If Ind T (a) ≠ 0, then Theorem 7.8 and the implication (i) ⇒ (iii) of Theorem 8.1 tell us that a^{−1} is a polynomial times a function in H^{∞} or \overline{H^{∞}}. As functions in H^{∞} or \overline{H^{∞}} cannot have jumps, this case is impossible.

We remark that Heinig and Hellinger [13] proved the equivalence (i) ⇔ (iii) of Theorem 8.2 for symbols in the Wiener algebra W. Corollary 8.3 was known to Silbermann and led him to the introduction of condition (ii). In the case of block Toeplitz matrices, (iii) and (ii) are no longer equivalent; Silbermann proved that then the validity of (15) for some n_{0} ≥ 1 implies that

{T_{n}(a)} is an exact Moore-Penrose sequence,  (49)

and he conjectures that (49) is even equivalent to (15) for some n_{0} ≥ 1 (see [25]). The proofs of [13] and [25] differ from the proof given above.

9. l^{p} versus l^{2}

As shown in the previous section, many l^{2} results can be derived with the help of Theorem 7.1, which reduces problems for {A_{n}} to questions about the infinite diagonal operator

diag (s_{1}^{(2)}(A_{1}), s_{1}^{(2)}(A_{2}), s_{2}^{(2)}(A_{2}), s_{1}^{(2)}(A_{3}), s_{2}^{(2)}(A_{3}), s_{3}^{(2)}(A_{3}), …).

It would therefore be very nice to have an analogous result for l^{p}. For example, one could ask the following: given A_{n} ∈ B(C_{p}^{n}), are there invertible isometries U_{n}, V_{n} ∈ B(C_{p}^{n}) and a diagonal matrix S_{n} ∈ B(C_{p}^{n}) such that A_{n} = U_{n}S_{n}V_{n} ? If the answer were “yes”, we had

Σ^{(p)}(A_{n}) = Σ^{(p)}(S_{n}),

and Theorem 11.11.3 of [16] would tell us that Σ^{(p)}(S_{n}) is the collection of the moduli of the diagonal elements of S_{n}.

However, the answer to the above question is “no”. The reason is the dramatic loss of symmetry when passing from l^{2} to l^{p}. Looking at the (real) unit spheres

S_{1}^{(p)} := {(x, y) ∈ R^{2} : |x|^{p} + |y|^{p} = 1},

we see that S_{1}^{(2)} has the symmetry group O(2), while the symmetry group of S_{1}^{(p)} (p ≠ 2) is the dieder group D_{4}, which contains only 8 elements. Equivalently, the invertible isometries in B(C_{2}^{2}) are the 2 × 2 unitary matrices, whereas a matrix U_{2} ∈ B(C_{p}^{2}) (p ≠ 2) is an invertible isometry if and only if

U_{2} = ( \begin{matrix}λ & 0 \\ 0 & μ \end{matrix} ) or U_{2} = ( \begin{matrix}0 & λ \\ μ & 0 \end{matrix} ) with (λ, μ) ∈ T^{2}.

Thus, a matrix A_{2} ∈ B(C_{p}^{2}) (p ≠ 2) is of the form A_{2} = U_{2}S_{2}V_{2} with invertible isometries U_{2}, V_{2} and a diagonal matrix S_{2} if and only if

A_{2} = ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) or A_{2} = ( \begin{matrix}0 & a \\ b & 0 \end{matrix} ) with (a, b) ∈ C^{2}.

I even suspect that relaxing the above question will not be successful.

Conjecture 9.1. Fix p ≠ 2 and let 1/p + 1/q = 1. There is no number M ∈ (1, ∞) with the following property: given any sequence {A_{n}} of matrices A_{n} ∈ B(C_{p}^{n}) such that sup ∥A_{n}∥_{p} < ∞ and sup ∥A_{n}∥_{q} < ∞, there are invertible matrices U_{n}, V_{n} ∈ B(C_{p}^{n}) and diagonal matrices S_{n} ∈ B(C_{p}^{n}) such that A_{n} = U_{n}S_{n}V_{n} and

∥U_{n}∥_{p} ≤ M, ∥U_{n}^{−1}∥_{p} ≤ M, ∥V_{n}∥_{p} ≤ M, ∥V_{n}^{−1}∥_{p} ≤ M

for all n.

Finally, for the reader’s convenience, we add a proof of (3).

Proposition 9.2. If A ∈ B(C_{p}^{n}), then s_{1}^{(p)}(A) = 1/∥A^{−1}∥_{p} if A is invertible and s_{1}^{(p)}(A) = 0 if A is not invertible.

Proof. Suppose A is not invertible. Then Ker A ≠ {0}. Let Z be any direct complement of Ker A in C_{p}^{n} and let P : C_{p}^{n} → Z be the projection onto Z parallel to Ker A. Clearly, P ∈ F_{n−1}^{(n)} and thus F := AP ∈ F_{n−1}^{(n)}. If x ∈ C^{n}, then x = x_{0} + x_{1} with x_{0} ∈ Ker A and x_{1} = P x ∈ Z. Therefore

(A − F)x = Ax − AP x = A(x_{0} + P x) − AP x = 0,

which implies that A − F = 0 and hence dist (A, F_{n−1}^{(n)}) = 0.

Now suppose A is invertible. We then have

∥A^{−1}∥_{p} = sup_{x≠0} \frac{∥A^{−1}x∥_{p}}{∥x∥_{p}} = sup_{z≠0} \frac{∥z∥_{p}}{∥Az∥_{p}} = ( inf_{z≠0} \frac{∥Az∥_{p}}{∥z∥_{p}} )^{−1},

whence

1/∥A^{−1}∥_{p} = inf_{z≠0} \frac{∥Az∥_{p}}{∥z∥_{p}} = min_{∥z∥_{p}=1} ∥Az∥_{p} =: ∥Ae_{0}∥_{p}  (50)

with some e_{0} ∈ C_{p}^{n} of norm 1. Put span {e_{0}} = {λe_{0} : λ ∈ C} and let X be any direct complement of span {e_{0}} in C_{p}^{n}. The functional

φ : span {e_{0}} → C, λe_{0} ↦ λ

clearly has the norm 1. By the Hahn-Banach theorem, there is a functional Φ : C_{p}^{n} → C such that Φ(λe_{0}) = λ and ∥Φ∥ = 1. Define F ∈ B(C_{p}^{n}) by F x := Ax − Φ(x)Ae_{0}. Since

F (λe_{0}) = λAe_{0} − λAe_{0} = 0,

we see that F ∈ F_{n−1}^{(n)}. Because

∥Ax − F x∥_{p} = ∥Φ(x)Ae_{0}∥_{p} = |Φ(x)| ∥Ae_{0}∥_{p} ≤ ∥x∥_{p} ∥Ae_{0}∥_{p},

it results that ∥A − F ∥_{p} ≤ ∥Ae_{0}∥_{p}. From (50) we therefore deduce that s_{1}^{(p)}(A) ≤ 1/∥A^{−1}∥_{p}.

To prove that s_{1}^{(p)}(A) ≥ 1/∥A^{−1}∥_{p}, let G be any matrix in F_{n−1}^{(n)}. If ∥I −A^{−1}G∥_{p} were less than 1, then A^{−1}G and thus G were invertible, which is impossible. Thus ∥I − A^{−1}G∥_{p} ≥ 1. We therefore have

1 ≤ ∥I − A^{−1}G∥_{p} = ∥A^{−1}(A − G)∥_{p} ≤ ∥A^{−1}∥_{p} ∥A − G∥_{p},

which implies that 1/∥A^{−1}∥_{p} ≤ ∥A − G∥_{p}. As G ∈ F_{n−1}^{(n)} was arbitrary, it follows that 1/∥A^{−1}∥_{p} ≤ s_{1}^{(p)}(A).

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Albrecht Böttcher

Fakultät für Mathematik TU Chemnitz-Zwickau 09107 Chemnitz Germany 30

Documenta Mathematica 2 (1997)