Documenta Mathematica Journal der Deutschen Mathematiker-Vereinigung Band 
 ISSN

-

Documenta Mathematica

Journal der Deutschen Mathematiker-Vereinigung

Band 2 1997

ISSN 1431-0635 Print ISSN 1431-0643Internet

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ISSN 1431-0635Documenta Mathematica(Print) ISSN 1431-0643Documenta Mathematica(Internet)

Anschrift des technischen geschaftsfuhrenden Herausgebers:

Ulf Rehmann, Fakultat fur Mathematik, Universitat Bielefeld, Postfach 100131, D-33501 Bielefeld Copyright c1997 fur das Layout: Ulf Rehmann

Documenta Mathematica

Journal der Deutschen Mathematiker-Vereinigung Band 2, 1997

A. Bottcher

On the Approximation Numbers

of Large Toeplitz Matrices 1{29

Amnon Besser

On the Finiteness of ^X for Motives

Associated to Modular Forms 31{46

A. Langer

Selmer Groups and Torsion Zero Cycles on the

Selfproduct of a Semistable Elliptic Curve 47{59 Christian Leis

Hopf-Bifurcation in Systems with Spherical Symmetry

Part I : Invariant Tori 61{113

Jane Arledge, Marcelo Laca and Iain Raeburn Semigroup Crossed Products and Hecke Algebras

Arising from Number Fields 115{138

Joachim Cuntz

Bivariante K-Theorie fur lokalkonvexe Algebren

und der Chern-Connes-Charakter 139{182

Henrik Kratz

Compact Complex Manifolds

with Numerically Effective Cotangent Bundles 183{193 Ekaterina Amerik

Maps onto Certain Fano Threefolds 195{211

Jonathan Arazy and Harald Upmeier Invariant Inner Product in Spaces of Holomorphic Functions

on Bounded Symmetric Domains 213{261

Victor Nistor

Higher Index Theorems and

the Boundary Map in Cyclic Cohomology 263{295 Oleg T. Izhboldin and Nikita A. Karpenko

On the Group H³(F( ;D)=F) 297{311

Udo Hertrich-Jeromin and Franz Pedit Remarks on the Darboux Transform of

Isothermic Surfaces 313{333

iii

Udo Hertrich-Jeromin

Supplement on Curved Flats in the Space of Point Pairs and Isothermic Surfaces:

A Quaternionic Calculus 335{350

Ernst-Ulrich Gekeler

On the Cuspidal Divisor Class Group of a

Drinfeld Modular Curve 351{374

Mikael Rrdam

Stability ofC-Algebras is Not a Stable Property 375{386

iv

Doc. Math. J. DMV 1

On the Approximation Numbers of Large Toeplitz Matrices

A. Bottcher Received: January 14, 1997 Communicated by Alfred K. Louis

Abstract. The kth approximation number s⁽_k^p)(An) of a complex nn matrix Anis dened as the distance of An to the nn matrices of rank at most n k. The distance is measured in the matrix norm associated with the l^p norm (1 < p < ¹) on

C

ⁿ. In the case p = 2, the approximation numbers coincide with the singular values.

We establish several properties of s⁽_k^p)(An) provided Anis the nn trunca- tion of an innite Toeplitz matrix A and n is large. As n^!1, the behavior of s⁽_k^p)(An) 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 l².

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

1. Introduction

Throughout this paper we tacitly identify a complex nn matrix with the operator it induces on

C

ⁿ. For 1 < p <¹, we denote by

C

_npthe space

C

ⁿwith the l^p norm,

kx^kp:=^jx1^jp+ ::: +^jxn^jp¹=p; and given a complex nn matrix An, we put

kAn^kp:= sup_x

6=0

kAnx^kp=^kx^kp

: (1)

Research supported by the Alfried Krupp Forderpreis fur junge Hochschullehrer of the Krupp Foundation

2 A. Bottcher

We let ^B(

C

_np) stand for the Banach algebra of all complex nn matrices with the norm (1). For j^2f0;1;:::;n^g, let^F_j⁽ⁿ⁾be the collection of all complex nn matrices of rank at most j, i.e., let

F

(n)

j :=ⁿF ^2B(

C

_np) : dimImFj^o:

The kth approximation number (k^2f0;1;:::;n^g) of An^2B(

C

_np) is dened as s⁽_k^p)(An) := dist(An;^F_{n k}⁽ⁿ⁾) := min^nkAn Fn^kp: Fn^2Fn k⁽n)

o: (2) (note that^F_j⁽ⁿ⁾is a closed subset of^B(

C

_np)). Clearly,

0 = s⁽₀^p)(A_n)s⁽₁^p)(A_n):::s⁽_n^p)(A_n) =^kA_n^k_p: It is easy to show (see Proposition 9.2) that

s⁽₁^p)(An) =

1=^kA_n^1kp if An is invertible;

0 if An is not invertible: (3) Notice also that in the case p = 2 the approximation numbers s⁽²⁾₁ (An);:::;s⁽²⁾n (An) are just the singular values of An, i.e., the eigenvalues of (A_nAn)¹⁼².

Let

T

be the complex unit circle and let a²L¹:= L¹(

T

). The nn Toeplitz matrix Tn(a) generated by a is the matrix

Tn(a) := (aj k)_nj;k=1 (4)

where a_l(l²

Z

) is the lth Fourier coecient of a, al:= 12^Z₀²a(eⁱ)e ^ild:

This paper is devoted to the limiting behavior of the numbers s⁽_k^p)(Tn(a)) as n goes to innity.

Of course, the study of properties of Tn(a) as n^!1leads to the consideration of the innite Toeplitz matrix

T(a) := (aj k)¹_j;k=1:

The latter matrix induces a bounded operator on l² := l²(

N

) if (and only if) a²L¹. Acting with T(a) on l^p:= l^p(

N

) is connected with a multiplier problem in case p⁶= 2.

We let Mpstand for the set of all a²L¹for which T(a) generates a bounded operator on l^p. The norm of this operator is denoted by ^kT(a)^kp. The function a is usually referred to as the symbol of T(a) and Tn(a).

In this paper, we prove the following results.

Theorem 1.1. If a²Mp then for eachk,

s⁽_{n k}^p) Tn(a)^!kT(a)^kp as n^!1:

Approximation Numbers of Toeplitz Matrices 3

Theorem 1.2. If a²Mp and T(a)is not normally solvable on l^p then for eachk, s⁽_k^p)Tn(a)^!0 as n^!1

Let M^h2ⁱ := L¹. For p⁶= 2, we dene M^hpⁱ as the set of all functions a²L¹ which belong to Mp~for all ~p in some open neighborhood of p (which may depend on a). A well known result by Stechkin says that a ² Mp for all p ²(1;¹) whenever a²L¹and the total variation V1(a) of a is nite and that in this case

kT(a)^k_pC_p^ka^k1+ V1(a) (5) with some constant Cp<¹(see, e.g., [5, Section 2.5(f)] for a proof). We denote by PC the closed subalgebra of L¹ constituted by all piecewise continuous functions.

Thus, a²PC if and only if a²L¹and the one-sided limits a(t0) := lim_"

!00a(e^i(+"))

exist for every t = eⁱ²

T

. By virtue of (5), the intersection PC^\M^hpⁱ contains all piecewise continuous functions of nite total variation.

Throughout what follows we dene q²(1;¹) by 1=p + 1=q = 1 and we put [p;q] :=^hmin^fp;q^g;max^fp;q^gi:

One can show that if a ² Mp, then a ² Mr for all r ² [p;q] (see, e.g., [5, Section 2.5(c)]).

Here is the main result of this paper.

Theorem 1.3. Leta be a function in PC^\M^h_pⁱ and suppose T(a) is Fredholm of the same index k (²

Z

)onl^r for allr²[p;q]. Then

nlim^!1s^(j_k^p)jTn(a)= 0 and liminf_n

!1

s^(j_k^p)j₊₁Tn(a)> 0:

For p = 2, Theorems 1.2 and 1.3 are special cases of results by Roch and Silber- mann [20], [21]. Since a Toeplitz operator on l² 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 ⁶= 2. If a ² C^\M^h_pⁱ, 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 a²PC^\M^hpⁱ we have three mutually excluding possibilities (see Section 3):

(i) T(a) is Fredholm of the same index k on l^r for all r²[p;q];

(ii) T(a) is not normally solvable on l^p or not normally solvable on l^q;

4 A. Bottcher

(iii) T(a) is normally solvable on l^p and l^q but not normally solvable on l^r for some r²(p;q) := [p;q]^nfp;q^g.

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)Tn(a)^!0 as n^!1 for every xed 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)(Tn(a)) is the history of the nite section method for Toeplitz operators: by virtue of (3), we have

s⁽₁^p)Tn(a)^!0^()kT_n¹(a)^kp^!1:

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

limsup_n

!1

kT_n¹(a)^kp<¹⁽⁾T(a)²0(l^p) (7) was proved by Gohberg and Feldman [7] in two cases: if a²C^\M^h_pⁱ(where C stands for the continuous functions on

T

) or if p = 2 and a²PC. For a²PC^\M^h_pⁱ, the equivalence

limsup_n

!1

kT_n¹(a)^kp<¹⁽⁾T(a)²0(l^r) for all r²[p;q] (8)

Approximation Numbers of Toeplitz Matrices 5

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 nitely many jumps, and nally 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 a²(C + H¹)^[(C + H¹)^[PQC

(see [4], [5]). Also notice that the implication \=⁾" of (8) is valid for every a²Mp. Treil [26] proved that there exist symbols a ² M^h2ⁱ = L¹ such that T(a)² 0(l²) but^kT_n¹(a)^k2is not uniformly bounded; concrete symbols with this property can be found in the recent article [2, Section 7.7].

The Toeplitz matrices Tn(') =

1 j k +

n

j;k=1 ( ⁶²

Z

)

are the elementary building blocks of general Toeplitz matrices with piecewise contin- uous symbols and have therefore been studied for some decades. The symbol is given by '(eⁱ) = sin eⁱe ⁱ; ²[0;2):

This is a function in PC with a single jump at eⁱ = 1. Tyrtyshnikov [27] focussed attention on the singular values of Tn('). He showed that

s⁽²⁾₁ Tn(')= O(1=n^{jj 1=2}) if ²

R

and ^jj> 1=2 and that there are constants c1;c2²(0;¹) such that

c1=logns⁽²⁾₁ T_n('1=2)c2=logn:

Curiously, the case ^jj< 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(')²0(l²)^()jRe^j< 1=2

(see, e.g., [7, Theorem IV.2.1] or [5, Proposition 6.24]), which together with (7) (for p = 2 and a²PC) implies that

liminf_n

!1

s⁽²⁾₁ Tn(')= 0 if ^jRe^j1=2 (9)

and liminf_n

!1

s⁽²⁾₁ Tn(')> 0 if ^jRe^j< 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

nlim^!1s⁽₁^p)Tn(')= 0 ^{() j}Re^jmin^f1=p;1=q^g

6 A. Bottcher

(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 < 1=q must be replaced by

1=q < Re < 1=p]).

As far as I know, collective phenomena of s⁽₁^p)(Tn(a));:::;s⁽n^p)(Tn(a)) have been studied only for p = 2, and throughout the rest of this section we abbreviate s⁽²⁾_k (Tn(a)) to sk(Tn(a)).

In 1920, Szego showed that if a²L¹ is real-valued and F is continuous on

R

,

then 1

n

n

X

k=1Fsk(Tn(a))^! 1 2

2

Z

0 F^ja(eⁱ)^jd: (10)

In the eighties, Parter [15] and Avram [1] extended this result to arbitrary (complex- valued) symbols a²L¹. Formula (10) implies that

nsk(Tn(a))^on_k=1 and ^nja(e^2ik=n)^jon_k=1 (11) are equally distributed (see [9] and [29]).

Research into the asymptotic distribution of the singular values of Toeplitz ma- trices was strongly motivated by a phenomenon discovered by C. Moler in the middle of the eighties. Moler observed that almost all singular values of Tn('1=2) are concen- trated in [ ";] where " is very small. Formula (10) provides a way to understand this phenomenon: letting F = 1 on [0; 2"] and F = 0 on [ ";] and taking into account that^j'1=2^j= 1, one gets

n1

n

X

k=1Fsk(Tn('1=2))^! 1 2

2

Z

0 F(1)d = F(1) = 0;

which shows that the percentage of the singular values of Tn('1=2) which are located in [0; 2"] goes to zero as n increases to innity.

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

a²L¹ and ^X

n²Z^jn^jjan^j2<¹ and that F ²C³(

R

), he showed that

n

X

k=1Fs²_k(Tn(a))= n2^Z₀²F^ja(eⁱ)^j2d + EF(a) + o(1)

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

(Tn(a)) :=ⁿs1(Tn(a));:::;sn(Tn(a))^o;

Approximation Numbers of Toeplitz Matrices 7

which, following the terminology of [19], are dened by part

(Tn(a)) := ^f2

R

: is partial limit of some sequence

f_n^g with _n²(T_n(a))^g; unif

(T_n(a)) := ^f2

R

: is the limit of some sequence

fn^g with n²(Tn(a))^g: It turned out that for large classes of symbols a we have

part

(Tn(a))= unif

(Tn(a))= spT(a)T(a)¹⁼² (12) where spA := ^{f 2}

C

: A I is not invertible^g denotes the spectrum of A (on l²) and a is dened by a(eⁱ) := a(eⁱ). Note that T(a) is nothing but the adjoint T(a) of T(a). Widom [32] proved (12) under the hypothesis that a²PC or that a is locally self-adjoint, while Silbermann [24] derived (12) for locally normal symbols.

Notice that symbols in PC or even in PQC are locally normal.

In the nineties, Tyrtyshnikov [28], [29] succeeded in proving that the sets (11) are equally distributed under the sole assumption that a²L²:= L²(

T

). His approach is based on the observation that if ^kAn Bn^kF = o(n), where ^kkF stands for the Frobenius (or Hilbert-Schmidt) norm, then Anand Bn have equally distributed singular values. The result mentioned can be shown by taking An = Tn(a) and choosing appropriate circulants for Bn.

The development received a new impetus from Heinig and Hellinger's 1994 paper [13]. They considered normally solvable Toeplitz operators on l² and studied the problem whether the Moore-Penrose inverses of T_n⁺(a) of Tn(a) converge strongly on l²to 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 ²C, then T(a) is normally solvable on l² if and only if a(t)⁶= 0 for all t ²

T

. When writing T_n⁺(a)^!T⁺(a), we actually mean that T_n⁺(a)Pn^!T⁺(a), where Pn

is the projection dened by

Pn:^fx1;x2;x3;:::^g7!fx1;x2;:::;xn;0;0;:::^g: (13) It is not dicult to verify that T_n⁺(a) ^!T⁺(a) strongly on l² if and only if T(a) is normally solvable and

limsup_n

!1

kT_n⁺(a)^k2<¹: (14) Heinig and Hellinger investigated normally solvable Toeplitz operators T(a) with symbols in the Wiener algebra W,

a²W ^()ka^kW :=^X

n²Z

jan^j<¹;

and they showed that then (14) is satised if and only if there is an n01 such that Ker T(a)ImPn⁰ and Ker T(a)ImPn⁰; (15)

8 A. Bottcher

where Ker A :=^fx²l²: Ax = 0^gand ImA :=^fAx : x²l^2g. (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 ^kT_n¹(a)^k2 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 ²spT(a), then

limsup_n

!1

kT_n⁺(a )^k2<¹ (16) can only hold if belongs to spTn(a) for all suciently large n. Consequently, (16) implies that lies in unif(sp Tn(a)), and the latter set is extremely \thin": it is contained in a nite union of analytic arcs (see [22] and [6]).

What has Moore-Penrose invertibility to do with singular values ? The answer is as follows: if An^2B(

C

ⁿ2) and sk(An) is the smallest nonzero singular value of An, then

kA⁺_n^k2= 1=sk(An):

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

(Tn(a))^f0^g[[d;¹) (17)

for all suciently large n.

Now Silbermann enters the scene. He replaced the Heinig-Hellinger problem by another one. Namely, given T(a), is there a sequence ^fBn^gof operators Bn^2B(

C

ⁿ₂) with the following properties: there exists a bounded operator B on l² such that

Bn^!B and B_n^!B strongly on l² and

kTn(a)BnTn(a) Tn(a)^k2^!0; ^kBnTn(a)Bn Bn^k2^!0;

k(BnTn(a)) BnTn(a)^k2^!0; ^k(Tn(a)Bn) Tn(a)Bn^k2^!0 ? Such a sequence ^fBn^gis 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 a²PC and T(a) is normally solvable, then T(a) always has an asymp- totic 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 cn^!0 and a number d > 0 such that

(Tn(a))[0;cn]^[[d;¹): (18) One says that (Tn(a)) has the splitting property if (18) holds with cn^!0 and d > 0.

Thus, Silbermann's result implies that if a²PC and T(a) is normally solvable on l², then (Tn(a)) has the splitting property.

Only recently, Roch and Silbermann [20], [21] were able to prove even much more. The sets (Tn(a)) are said to have the k-splitting property, where k0 is an integer, if (18) is true for some sequence cn ^!0 and some d > 0 and, in addition, exactly k singular values lie in [0;cn] and n k singular values are located in [d;¹)

Approximation Numbers of Toeplitz Matrices 9

(here multiplicities are taken into account). Equivalently, (Tn(a)) has the k-splitting property if and only if

nlim^!1s_k(T_n(a)) = 0 and liminf_n

!1

s_k+1(T_n(a)) > 0: (19) A normally solvable Toeplitz operator T(a) on l² with a symbol a ² PC is automatically Fredholm and therefore has some index k²

Z

. Roch and Silbermann [20], [21] discovered that then (Tn(a)) has the^jk^j-splitting property. In other words, if a²PC and T(a)²k(l²) then (19) holds with k replaced by^jk^j. 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 ^jk^j-splitting property must have its root in the possibility of \ignoring^jk^jdimensions" which led me to the observation that none of the works cited in this section makes use of the fact that sk(An) may alternatively be dened 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 l² and C-algebras to l^pand Banach algebras.

3. Toeplitz operators on l^p

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

Let Mp and M^hpⁱ be as in Section 1. The set Mp can be shown to be a Banach algebra with pointwise algebraic operations and the norm ^ka^kMp := ^kT(a)^kp. It is also well known that

Mp= MqM2= L¹ and

ka^k_Mp =^ka^k_Mq^ka^k_M² =^ka^k1 (20) (see, e.g., [5, Section 2.5]). We remark that working with M^hpⁱinstead of Mpis caused by the need of somehow reversing the estimate in (20). Suppose, for instance, p > 2 and a ²M^hpⁱ. Then a ²Mp+" for some " > 0, and the Riesz-Thorin interpolation theorem gives

ka^kMp ^ka^k_M^2ka^k1_p+"=^ka^k1ka^k1_Mp⁺_" (21) with some ²(0;1) depending only on p and ". The^ka^kMp⁺"on the right of (21) may in turn be estimated by Cp(^ka^k1+ V1(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, ImA, is a closed subset of l^p. The operator A is called Fredholm if it is normally solvable and the spaces

Ker A :=^fx²l^p: Ax = 0^g and CokerA := l^p=ImA have nite dimensions. In that case the index IndA is dened as

IndA := dimKer A dimCokerA:

10 A. Bottcher

We denote by (l^p) the collection of all Fredholm operators on l^p and by k(l^p) the operators in (l^p) whose index is k. The following four theorems are well known.

Comments are at the end of this section.

Theorem 3.1. Leta²Mp.

(a)Ifadoes not vanish identically, then the kernel ofT(a)onl^p or the kernel ofT(a) on l^q is trivial.

(b) The operatorT(a) is invertible onl^p if and only if T(a)²0(l^p). Of course, part (b) is a simple consequence of part (a).

Theorem 3.2. Let a²C^\M^hpⁱ. Then T(a) is normally solvable on l^p if and only if a(t)⁶= 0 for allt²

T

. In that case T(a)²(l^p)and

IndT(a) = winda;

where windais the winding number ofa about the origin.

Now let a²PC; t²

T

, and suppose a(t 0)⁶= a(t + 0). We denote by

Ap(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^f2=p;2=q^gand which lies on the right of the straight line passing rst a(t 0) and then a(t + 0) if 1 < p < 2 and on the left of this line if 2 < p <¹. 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 lling in the arcs ^Ap(a(t 0);a(t + 0)) for each jump.

In case this curve does not pass through the origin, we let winda^#_p be its winding number.

Theorem 3.3. Leta²PC^\M^hpⁱ. ThenT(a)is normally solvable on l^p if and only if 0⁶²a^#_p. In that case T(a)²(l^p)and

IndT(a) = winda^#_p: For a²PC and t²

T

, put

Op

a(t 0); a(t + 0):= ^[

r²[p;q]

Ar

a(t 0); a(t + 0): (22) If a(t 0)⁶= a(t + 0) and p⁶= 2, then^O_p(a(t 0);a(t + 0)) is a certain lentiform set.

Also for a²PC, let

a^#_[p;q] := ^[

r²[p;q]a^#_r :

Approximation Numbers of Toeplitz Matrices 11

Thus, a^#_[p;q] results from^R(a) by lling in the sets (22) between the endpoints of the jumps. If 0⁶²a^#_[p;q], then necessarily 0 ⁶²a^#₂ and we dene winda^#_[p;q] as winda^#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') 0⁶²a^#_[p;q] and winda^#_[p;q] = k;

(ii') 0²a^#_p ^[a^#_q;

(iii') 0²a^#_[p;q]ⁿ(a^#_p ^[a^#_q ).

For a ²Mp, let Tn(a) ^{2 B}(

C

_np) be the operator given by the matrix (4). One says that the sequence ^fTn(a)^g:=^fTn(a)^g1_n=1is stable if

limsup_n

!1

kT_n¹(a)^k_p<¹: Here we follow the practice of putting

kT_n¹(a)^kp =¹ if Tn(a) is not invertible.

In other words, ^fTn(a)^gis stable if and only if Tn(a) is invertible for all nn0 and there exists a constant M <¹ such that^kT_n¹(a)^kp M for all nn0. From (3) we infer that

fT_n(a)^g is stable ⁽⁾liminf_n

!1

s⁽₁^p)(T_n(a)) > 0:

Theorem 3.4. (a)If a²C^\M^h_pⁱ then

fT_n(a)^g is stable ⁽⁾ 0⁶²a(

T

) and winda = 0:

(b) If a²PC^\M^h_pⁱ then

fTn(a)^g is stable ⁽⁾ 0⁶²a^#_[p;q] and winda^#_[p;q]= 0:

As already said, these theorems are well known. Theorem 3.1 is due to Coburn (p = 2) and Duduchava (p ⁶= 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 p⁶= 2, Theorem 3.4 goes back to Gohberg and Feldman for a²C^\M^hpⁱ (general p) and a ² PC (p = 2), and it was obtained in the work of Verbitsky, Krupnik, Silbermann, and the author for a²PC^\M^hpⁱ and p⁶= 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, Theo- rem 3.4(b) may also be stated as follows: ^fTn(a)^g contains a stable subsequence

12 A. Bottcher

fTnj(a)^g(nj ^!1) if and only if 0⁶²a^#_[p;q] and winda^#_[p;q] = 0. Hence, we arrive at the conclusion that if a²PC^\M^h_pⁱ, then

s⁽₁^p)(Tn(a))^!0

()fTn(a)^g is stable

()0²a^#_[p;q] or 0⁶²a^#_[p;q] and winda^#_[p;q] ⁶= 0:

At this point the question of whether the lowest approximation number of Tn(a) goes to zero or not is completely disposed of for symbols a²PC^\M^h_pⁱ.

4. Proof of Theorem 1.1.

Contrary to what we want, let us assume that there is a c < ^kT(a)^kp such that s⁽_{n k}^p) (Tn(a)) c for all n in some innite set ^N. Since s⁽_{n k}^p) (Tn(a)) = dist(T_n(a);^F_k⁽ⁿ⁾), we can nd F_n^{2 F}_k⁽ⁿ⁾(n ^2N) so that ^kT_n(a) F_n^k_p c. For x = (x1;:::;x_n) and y = (y1;:::;y_n), we dene

(x;y) := x1y1+ :::+ xnyn: (23) By [16, Lemma B.4.11], there exist e⁽_jⁿ⁾²

C

_np; f_j⁽ⁿ⁾²

C

_np; _j⁽ⁿ⁾²

C

such that

Fnx =^Xk

j=1⁽_jⁿ⁾x;f_j⁽ⁿ⁾e⁽_jⁿ⁾ (x²

C

np);

ke⁽_j^n)kp= 1; ^kf_j^(n)kq = 1, and

j_j^(n)jkFn^kp^kTn(a)^kp+^kFn Tn(a)^kp^kT(a)^kp+ c (24) for all j^2f1;:::;k^g.

Fix x²

C

_np;y²

C

_nq and suppose^kx^kp= 1; ^ky^kq= 1. We then have

Tn(a)x;y ^Xk

j=1_j⁽ⁿ⁾x;f_j⁽ⁿ⁾e⁽_jⁿ⁾;y^kTn(a) Fn^kpc: (25) Clearly, (Tn(a)x;y)^!(T(a)x;y). From (24) and the Bolzano-Weierstrass theorem we infer that the sequence ^f(⁽₁ⁿ⁾;:::;_k⁽ⁿ⁾)^gn^2N has a converging subsequence. Without loss of generality suppose the sequence itself converges, i.e.

₁⁽ⁿ⁾;:::;_k^(n)!(1;:::;k)²

C

^k

as n ^{2 N} goes to innity. The vectors e⁽_jⁿ⁾ 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]), ^fe⁽_j^n)gn^2N and ^ff_j^(n)gn^2N have subsequences converging in the weak -topology. Again we may without loss of generality assume that

e⁽_j^n)!ej ²l^p; f_j^(n)!fj ²l^q

Approximation Numbers of Toeplitz Matrices 13

in the weak-topology as n^2N goes to innity.

From (25) we now obtain that if x ² l^p and y ² l^q have nite support and

kx^kp= 1; ^ky^kq = 1, then

T(a)x;y) ^Xk

j=1j(x;fj)(ej;y)c:

This implies that

kT(a) F^kpc (26)

where F is the nite-rank operator given by Fx :=^Xk

j=1j(x;fj)ej (x²l^p): (27) Let ^kT(a)^k(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),

kT(a)^k(ess)_p ^kT(a) F^kpc <^kT(a)^kp:

However, one always has ^kT(a)^k(ess)p = ^kT(a)^kp (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. LetAbe a bounded linear operator on l^p. (a)The operator Ais normally solvable on l^p if and only if

kA:=_x sup

2l^p;^kx^kp=1dist(x;KerA) <¹:

(b) If M is a closed subspace ofl^p and dim(l^p=M) <¹, then the normal solv- ability ofA^jM : 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. IfM is ak-dimensional subspace of

C

_np, then there exists a projection :

C

_np^!

C

_np such thatIm = M and^kkpk.

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

Theorem 1.2 is trivial in case a vanishes identically. So suppose a²Mp^nf0^gand T(a) is not normally solvable on l^p. Then the adjoint operator T(a) is not normally solvable on l^q. By Theorem 3.1(a), KerT(a) =^f0^gon l^p or Ker T(a) = ^f0^gon l^q.