## Documenta Mathematica

### Journal der Deutschen Mathematiker-Vereinigung

### Band 2 1997

ISSN 1431-0635 Print ISSN 1431-0643Internet

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Copyright c^{1997 f¨}ur das Layout: Ulf Rehmann

### Documenta Mathematica

Journal der Deutschen Mathematiker-Vereinigung Band 2, 1997

A. B¨ottcher

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 f¨ur 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 H^{3}(F(ψ, D)/F) 297–311

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

Isothermic Surfaces 313–333

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 Rørdam

Stability ofC^{∗}-Algebras is Not a Stable Property 375–386

Doc. Math. J. DMV 1

### On the Approximation Numbers of Large Toeplitz Matrices

A. B¨ottcher^{∗}

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

Abstract. The kth approximation number s^{(p)}_{k} (A_{n}) of a complex n×n
matrixA_{n} is deﬁned as the distance ofA_{n} to then×nmatrices 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 ofs^{(p)}_{k} (A_{n}) providedA_{n}is then×ntrunca-
tion of an inﬁnite Toeplitz matrixAandnis large. Asn→ ∞, the behavior
of s^{(p)}_{k} (A_{n}) depends heavily on the Fredholm properties (and, in particular,
on the index) ofAonl^{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 onl^{2}.

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

1. Introduction

Throughout this paper we tacitly identify a complexn×nmatrix with the operator
it induces onC^{n}. For 1< p <∞, we denote by C^{n}_{p} the space C^{n} with thel^{p} norm,

kxkp:=

|x_{1}|^{p}+. . .+|x_{n}|^{p}1/p

,
and given a complex n×nmatrixA_{n}, we put

kAnkp:= sup

x6=0

kAnxkp/kxkp

. (1)

∗Research supported by the Alfried Krupp F¨orderpreis f¨ur junge Hochschullehrer of the Krupp Foundation

2 A. B¨ottcher

We let B(C^{n}_{p}) stand for the Banach algebra of all complexn×n matrices with the
norm (1). Forj∈ {0,1, . . . , n}, letFj^{(n)}be the collection of all complexn×nmatrices
of rank at mostj, i.e., let

Fj^{(n)}:=n

F ∈ B(C^{n}_{p}) : dim ImF≤jo
.

Thekth approximation number (k∈ {0,1, . . . , n}) ofAn∈ B(C^{n}_{p}) is deﬁned as
s^{(p)}_{k} (A_{n}) := dist (A_{n},Fn−k^{(n)}) := minn

kA_{n}−F_{n}kp:F_{n} ∈ Fn−k^{(n)}

o. (2)

(note thatF_{j}^{(n)}is a closed subset ofB(C^{n}_{p})). Clearly,

0 =s^{(p)}_{0} (An)≤s^{(p)}_{1} (An)≤. . .≤s^{(p)}_{n} (An) =kAnkp.
It is easy to show (see Proposition 9.2) that

s^{(p)}_{1} (A_{n}) =

1/kA^{−1}_{n} kp if A_{n} is invertible,

0 if A_{n} is not invertible. (3)

Notice also that in the casep= 2 the approximation numberss^{(2)}_{1} (A_{n}), . . . , s^{(2)}_{n} (A_{n})
are just the singular values ofA_{n}, i.e., the eigenvalues of (A^{∗}_{n}A_{n})^{1/2}.

LetTbe the complex unit circle and leta∈L^{∞}:=L^{∞}(T). Then×nToeplitz
matrixT_{n}(a) generated byais the matrix

T_{n}(a) := (aj−k)^{n}_{j,k=1} (4)

where al(l∈Z) is thelth Fourier coeﬃcient ofa,
a_{l}:= 1

2π Z2π 0

a(e^{iθ})e^{−ilθ}dθ.

This paper is devoted to the limiting behavior of the numberss^{(p)}_{k} (Tn(a)) as ngoes
to inﬁnity.

Of course, the study of properties ofTn(a) asn→ ∞leads to the consideration of the inﬁnite Toeplitz matrix

T(a) := (aj−k)^{∞}_{j,k=1}.

The latter matrix induces a bounded operator onl^{2} :=l^{2}(N) if (and only if)a∈L^{∞}.
Acting withT(a) onl^{p}:=l^{p}(N) is connected with a multiplier problem in casep6= 2.

We letM_{p}stand for the set of alla∈L^{∞}for whichT(a) generates a bounded operator
on l^{p}. The norm of this operator is denoted by kT(a)kp. The function ais usually
referred to as the symbol of T(a) andTn(a).

In this paper, we prove the following results.

Theorem 1.1. If a∈M_{p} then for eachk,
s^{(p)}_{n−k}

T_{n}(a)

→ kT(a)kp as n→ ∞.

Approximation Numbers of Toeplitz Matrices 3

Theorem 1.2. If a∈Mp andT(a)is not normally solvable onl^{p} then for eachk,
s^{(p)}_{k}

T_{n}(a)

→0 as n→ ∞

Let M_{h2i} :=L^{∞}. Forp6= 2, we deﬁne M_{hpi} as the set of all functions a∈L^{∞}
which belong toM_{p}_{˜}for all ˜pin some open neighborhood ofp(which may depend on
a). A well known result by Stechkin says that a∈ M_{p} for allp ∈(1,∞) whenever
a∈L^{∞}and the total variationV_{1}(a) ofais ﬁnite and that in this case

kT(a)kp≤Cp

kak∞+V1(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 Cif and only if a∈L^{∞}and the one-sided limits
a(t±0) := lim

ε→0±0a(e^{i(θ+ε)})

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

Throughout what follows we deﬁneq∈(1,∞) by 1/p+ 1/q= 1 and we put [p, q] :=h

min{p, q},max{p, q}i .

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∩Mhpi and suppose T(a)is Fredholm of
the same index −k(∈Z)onl^{r} for allr∈[p, q]. Then

n→∞lim s^{(p)}_{|k|}

T_{n}(a)

= 0 and lim inf

n→∞ s^{(p)}_{|k|+1}
T_{n}(a)

>0.

Forp= 2, Theorems 1.2 and 1.3 are special cases of results by Roch and Silber-
mann [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 casep= 2.

Now suppose p 6= 2. If a ∈ C∩Mhpi, 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_{hpi} we have
three mutually excluding possibilities (see Section 3):

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

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

4 A. B¨ottcher

(iii) T(a) is normally solvable onl^{p} andl^{q} but not normally solvable onl^{r} for some
r∈(p, q) := [p, q]\ {p, q}.

In the case (i) we can apply Theorem 1.3. Since
s^{(p)}_{k}

Tn(a)

=s^{(q)}_{k}
Tn(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^{(p)}_{k}

Tn(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 numbers^{(p)}_{1} (T_{n}(a)) is the history of the ﬁnite
section method for Toeplitz operators: by virtue of (3), we have

s^{(p)}_{1}
Tn(a)

→0⇐⇒ kT_{n}^{−1}(a)kp→ ∞.

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

lim sup

n→∞ kT_{n}^{−1}(a)kp<∞ ⇐⇒T(a)∈Φ_{0}(l^{p}) (7)
was proved by Gohberg and Feldman [7] in two cases: ifa∈C∩Mhpi(whereCstands
for the continuous functions onT) or if p= 2 anda∈P C. Fora∈P C∩Mhpi, the
equivalence

lim sup

n→∞ kT_{n}^{−1}(a)kp<∞ ⇐⇒T(a)∈Φ_{0}(l^{r}) for all r∈[p, q] (8)

Approximation Numbers of Toeplitz Matrices 5

holds. This was shown by Verbitsky and Krupnik [30] in the case whereahas 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+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_{h2i} = L^{∞} such thatT(a)∈ Φ_{0}(l^{2})
butkT_{n}^{−1}(a)k2is 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}) =

1 j−k+γ

n j,k=1

(γ6∈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

ϕγ(e^{iθ}) = π

sinπγe^{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 ofT_{n}(ϕ_{γ}). He showed that

s^{(2)}_{1}

T_{n}(ϕ_{γ})

=O(1/n^{|γ|−1/2}) if γ∈R and |γ|>1/2
and that there are constantsc1, c2∈(0,∞) such that

c1/logn≤s^{(2)}_{1}

Tn(ϕ_{1/2})

≤c2/logn.

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})⇐⇒ |Reγ|<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^{(2)}_{1}

T_{n}(ϕ_{γ})

= 0 if |Reγ| ≥1/2 (9)

and

lim inf

n→∞ s^{(2)}_{1}

T_{n}(ϕ_{γ})

>0 if |Reγ|<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

n→∞lim s^{(p)}_{1}

T_{n}(ϕ_{γ})

= 0 ⇐⇒ |Reγ| ≥min{1/p,1/q}

6 A. B¨ottcher

(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 < Reβ < 1/q must be replaced by

−1/q <Reβ <1/p]).

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

In 1920, Szeg¨o showed that ifa∈L^{∞} is real-valued andF is continuous onR,
then

1 n

Xn k=1

F

s_{k}(T_{n}(a))

→ 1 2π

Z2π 0

F

|a(e^{iθ})|

dθ. (10)

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

ns_{k}(T_{n}(a))on

k=1 and n

|a(e^{2πik/n})|on

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 ofT_{n}(ϕ_{1/2}) are concen-
trated in [π−ε, π] whereεis very small. Formula (10) provides a way to understand
this phenomenon: lettingF = 1 on [0, π−2ε] andF = 0 on [π−ε, π] and taking into
account that|ϕ_{1/2}|= 1, one gets

1 n

Xn k=1

F

sk(Tn(ϕ_{1/2}))

→ 1 2π

Z2π 0

F(1)dθ=F(1) = 0,

which shows that the percentage of the singular values ofT_{n}(ϕ_{1/2}) which are located
in [0, π−2ε] goes to zero asnincreases 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 X

n∈Z

|n| |a_{n}|^{2}<∞
and thatF ∈C^{3}(R), he showed that

Xn k=1

F

s^{2}_{k}(T_{n}(a))

= n 2π

Z2π 0

F

|a(e^{iθ})|^{2}

dθ+E_{F}(a) +o(1)

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

Σ(T_{n}(a)) :=n

s_{1}(T_{n}(a)), . . . , s_{n}(T_{n}(a))o
,

Approximation Numbers of Toeplitz Matrices 7

which, following the terminology of [19], are deﬁned by
Λ_{part}

Σ(T_{n}(a))

:= {λ∈R:λ is partial limit of some sequence {λn} with λn∈Σ(Tn(a))}, Λunif

Σ(Tn(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 symbolsawe have

Λpart

Σ(Tn(a))

= Λunif

Σ(Tn(a))

= sp

T(a)T(a)1/2

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

Notice that symbols inP C or even inP 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 kA_{n} −B_{n}kF = o(n), where k · kF 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 ofT_{n}^{+}(a) ofT_{n}(a) converge strongly on
l^{2}to the Moore-Penrose inverseT^{+}(a) ofT(a). Recall that the Moore-Penrose inverse
of a normally solvable Hilbert space operatorAis 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 ifa(t)6= 0 for all t ∈T.

When writingT_{n}^{+}(a)→T^{+}(a), we actually mean thatT_{n}^{+}(a)P_{n}→T^{+}(a), where P_{n}
is the projection deﬁned by

P_{n}:{x_{1}, x_{2}, x_{3}, . . .} 7→ {x_{1}, x_{2}, . . . , x_{n},0,0, . . .}. (13)
It is not diﬃcult to verify that T_{n}^{+}(a)→T^{+}(a) strongly onl^{2} if and only ifT(a) is
normally solvable and

lim sup

n→∞ kT_{n}^{+}(a)k2<∞. (14)

Heinig and Hellinger investigated normally solvable Toeplitz operatorsT(a) with symbols in the Wiener algebraW,

a∈W ⇐⇒ kakW :=X

n∈Z

|a_{n}|<∞,

and they showed that then (14) is satisﬁed if and only if there is ann_{0}≥1 such that
KerT(a)⊂ImP_{n}_{0} and KerT(a)⊂ImP_{n}_{0}, (15)

8 A. B¨ottcher

where KerA:={x∈l^{2}: Ax= 0}and ImA:={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 kT_{n}^{−1}(a)k2 is uniformly bounded. The
really interesting case is the one in whichT(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 onT) andλ∈spT(a), then

lim sup

n→∞ kT_{n}^{+}(a−λ)k2<∞ (16)

can only hold if λbelongs to spT_{n}(a) for all suﬃciently largen. Consequently, (16)
implies that λ lies in Λ_{unif}(spT_{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: ifA_{n}∈ B(C^{n}_{2}) ands_{k}(A_{n}) is the smallest nonzero singular value ofA_{n},
then

kA^{+}_{n}k2= 1/sk(An).

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

Σ(Tn(a))⊂ {0} ∪[d,∞) (17)

for all suﬃciently large n.

Now Silbermann enters the scene. He replaced the Heinig-Hellinger problem by
another one. Namely, givenT(a), is there a sequence {Bn}of operatorsBn ∈ B(C^{n}_{2})
with the following properties: there exists a bounded operator B onl^{2} such that

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

kT_{n}(a)B_{n}T_{n}(a)−T_{n}(a)k2→0, kB_{n}T_{n}(a)B_{n}−B_{n}k2→0,
k(BnTn(a))^{∗}−BnTn(a)k2→0, k(Tn(a)Bn)^{∗}−Tn(a)Bnk2→0 ?
Such a sequence {B_{n}}is referred to as an asymptotic Moore-Penrose inverse ofT(a).

In view of the (instable) conditions (15), the following result by Silbermann [25] is surprising: ifa∈P CandT(a) is normally solvable, thenT(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

Σ(Tn(a))⊂[0, cn]∪[d,∞). (18) One says that Σ(Tn(a)) has the splitting property if (18) holds withcn→0 andd >0.

Thus, Silbermann’s result implies that ifa∈P CandT(a) is normally solvable onl^{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(Tn(a)) are said to have thek-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}] andn−k singular values are located in [d,∞)

Approximation Numbers of Toeplitz Matrices 9

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

n→∞lim sk(Tn(a)) = 0 and lim inf

n→∞ sk+1(Tn(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 indexk∈Z. Roch and Silbermann
[20], [21] discovered that then Σ(Tn(a)) has the|k|-splitting property. In other words,
ifa∈P Cand T(a)∈Φk(l^{2}) then (19) holds withkreplaced by|k|. Notice that this
Theorem 1.3 forp= 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
thats_{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 froml^{2} andC^{∗}-algebras tol^{p}and Banach algebras.

3. Toeplitz operators on l^{p}

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

LetMp andMhpi be as in Section 1. The set Mp can be shown to be a Banach algebra with pointwise algebraic operations and the norm kakMp := kT(a)kp. It is also well known that

M_{p}=M_{q}⊂M_{2}=L^{∞}
and

kakMp=kakMq≥ kakM2 =kak∞ (20)
(see, e.g., [5, Section 2.5]). We remark that working withMhpiinstead ofM_{p}is caused
by the need of somehow reversing the estimate in (20). Suppose, for instance, p >2
and a ∈Mhpi. Then a ∈M_{p+ε} for some ε >0, and the Riesz-Thorin interpolation
theorem gives

kakMp≤ kak^{γ}M2kak^{1−γ}p+ε =kak^{γ}∞kak^{1−γ}Mp+ε (21)
with someγ∈(0,1) depending only onpandε. ThekakMp+εon the right of (21) may
in turn be estimated by C_{p}(kak∞+V_{1}(a)) (recall Stechkin’s inequality (5)) provided
ahas bounded total variation.

A bounded linear operator A onl^{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

KerA:={x∈l^{p}:Ax= 0} and CokerA:=l^{p}/ImA
have ﬁnite dimensions. In that case the index IndAis deﬁned as

IndA:= dim KerA−dim CokerA.

10 A. B¨ottcher

We denote by Φ(l^{p}) the collection of all Fredholm operators onl^{p} and by Φk(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. Leta∈Mp.

(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)∈Φ_{0}(l^{p}).

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

Theorem 3.2. Let a∈C∩M_{hpi}. Then T(a)is normally solvable on l^{p} if and only
if a(t)6= 0 for allt∈T. In that case T(a)∈Φ(l^{p})and

IndT(a) =−winda, where windais the winding number ofa about the origin.

Now leta∈P C, t∈T, and supposea(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{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,Ap(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) rangeR(a) ofaby ﬁlling 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. Leta∈P C∩M_{hpi}. ThenT(a)is normally solvable onl^{p} if and only
if 06∈a^{#}_{p}. In that case T(a)∈Φ(l^{p})and

IndT(a) =−winda^{#}_{p}.

Fora∈P Candt∈T, put Op

a(t−0), a(t+ 0)

:= [

r∈[p,q]

Ar

a(t−0), a(t+ 0)

. (22)

If a(t−0)6=a(t+ 0) andp6= 2, thenOp(a(t−0), a(t+ 0)) is a certain lentiform set.

Also fora∈P C, let

a^{#}_{[p,q]} := [

r∈[p,q]

a^{#}_{r}.

Approximation Numbers of Toeplitz Matrices 11

Thus, a^{#}_{[p,q]} results fromR(a) by ﬁlling in the sets (22) between the endpoints of the
jumps. If 06∈a^{#}_{[p,q]}, then necessarily 0 6∈a^{#}_{2} and we deﬁne 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’) 06∈a^{#}_{[p,q]} and winda^{#}_{[p,q]} =k;

(ii’) 0∈a^{#}_{p} ∪a^{#}_{q};

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

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

lim sup

n→∞ kT_{n}^{−1}(a)kp<∞.
Here we follow the practice of putting

kT_{n}^{−1}(a)kp =∞ if T_{n}(a) is not invertible.

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

{Tn(a)} is stable ⇐⇒lim inf

n→∞ s^{(p)}_{1} (Tn(a))>0.

Theorem 3.4. (a)If a∈C∩M_{hpi} then

{Tn(a)} is stable ⇐⇒ 06∈a(T) and winda= 0.

(b) If a∈P C∩Mhpi then

{T_{n}(a)} is stable ⇐⇒ 06∈a^{#}_{[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 casep= 2 and was established by Duduchava
forp6= 2, Theorem 3.4 goes back to Gohberg and Feldman fora∈C∩M_{hpi} (general
p) and a ∈ P C (p = 2), and it was obtained in the work of Verbitsky, Krupnik,
Silbermann, and the author fora∈P C∩M_{hpi} andp6= 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: {T_{n}(a)} contains a stable subsequence

12 A. B¨ottcher

{Tnj(a)}(nj → ∞) if and only if 06∈a^{#}_{[p,q]} and winda^{#}_{[p,q]} = 0. Hence, we arrive at
the conclusion that ifa∈P C∩M_{hpi}, then

s^{(p)}_{1} (Tn(a))→0

⇐⇒ {T_{n}(a)} is stable

⇐⇒0∈a^{#}_{[p,q]} or

06∈a^{#}_{[p,q]} and winda^{#}_{[p,q]} 6= 0
.

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

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^{(p)}_{n−k}(T_{n}(a)) ≤ c for all n in some inﬁnite set N. Since s^{(p)}_{n−k}(T_{n}(a)) =
dist (Tn(a),F_{k}^{(n)}), we can ﬁnd Fn ∈ F_{k}^{(n)}(n ∈ N) so that kTn(a)−Fnkp ≤ c. For
x= (x1, . . . , xn) andy= (y1, . . . , yn), we deﬁne

(x, y) :=x_{1}y_{1}+. . .+x_{n}y_{n}. (23)
By [16, Lemma B.4.11], there exist e^{(n)}_{j} ∈C^{n}_{p}, f_{j}^{(n)}∈C^{n}_{p}, γ_{j}^{(n)}∈Csuch that

Fnx= Xk j=1

γ^{(n)}_{j}

x, f_{j}^{(n)}

e^{(n)}_{j} (x∈C^{n}_{p}),
ke^{(n)}_{j} kp= 1, kf_{j}^{(n)}kq= 1, and

|γ_{j}^{(n)}| ≤ kF_{n}kp≤ kT_{n}(a)kp+kF_{n}−T_{n}(a)kp≤ kT(a)kp+c (24)
for allj∈ {1, . . . , k}.

Fixx∈C^{n}_{p}, y∈C^{n}_{q} and supposekxkp= 1, kykq= 1. We then have

Tn(a)x, y

− Xk j=1

γ_{j}^{(n)}

x, f_{j}^{(n)}

e^{(n)}_{j} , y≤ kTn(a)−Fnkp≤c. (25)
Clearly, (Tn(a)x, y)→(T(a)x, y). From (24) and the Bolzano-Weierstrass theorem we
infer that the sequence {(γ^{(n)}_{1} , . . . , 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)}

→(γ1, . . . , γk)∈C^{k}

as n ∈ N goes to inﬁnity. The vectors e^{(n)}_{j} 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^{(n)}_{j} }^{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^{(n)}_{j} →e_{j} ∈l^{p}, f_{j}^{(n)}→f_{j} ∈l^{q}

Approximation Numbers of Toeplitz Matrices 13

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

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

T(a)x, y)

− Xk j=1

γ_{j}(x, f_{j})(e_{j}, y)≤c.

This implies that

kT(a)−Fkp≤c (26) where F is the ﬁnite-rank operator given by

F x:=

Xk j=1

γj(x, fj)ej (x∈l^{p}). (27)
Let kT(a)k^{(ess)} denote the essential norm of T(a) on l^{p}, i.e. the distance ofT(a) to
the compact operators onl^{p}. By (26) and (27),

kT(a)k^{(ess)}p ≤ kT(a)−Fkp≤c <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. Let Abe a bounded linear operator on l^{p}.
(a)The operator Ais normally solvable on l^{p} if and only if

kA:= sup

x∈l^{p},kxkp=1

dist (x,KerA)<∞.

(b) If M is a closed subspace of l^{p} and dim (l^{p}/M)<∞, then the normal solv-
ability ofA|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. IfMis ak-dimensional subspace ofC^{n}_{p}, then there exists a projection
Π :C^{n}_{p} →C^{n}_{p} such thatIm Π =M andkΠkp≤k.

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

Theorem 1.2 is trivial in caseavanishes identically. So supposea∈Mp\ {0}and
T(a) is not normally solvable onl^{p}. Then the adjoint operatorT(a) is not normally
solvable on l^{q}. By Theorem 3.1(a), KerT(a) ={0}on l^{p} or KerT(a) = {0}on l^{q}.

14 A. B¨ottcher

Since s^{(p)}_{k} (T_{n}(a)) =s^{(q)}_{k} (T_{n}(a)), we may a priori assume that KerT(a) = {0}on l^{p}.
Abbreviate T(a) andT_{n}(a) toAandA_{n}, respectively.

Deﬁne P_{n} onl^{p} by (13) and let

V :=l^{p}→l^{p}, {x_{1}, x_{2}, x_{3}, . . .} 7→ {0, x_{1}, x_{2}, x_{3}, . . .}.

As A|ImV^{n} : ImV^{n} →l^{p} has the same matrix as AV^{n} : l^{p} → l^{p}, we deduce from
Theorem 5.1(b) that there is non≥0 such thatAV^{n}is normally solvable. Note that
Ker (AV^{n}) ={0}for alln≥0.

Letl^{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 whichx_{j}= 0 whenever j ≤n_{1} orj > n_{2}.

Lemma 5.3. There are0 =N0< N1< N2< . . .andzj ∈l^{p}(Nj−1, Nj] (j ≥1) such
that

kzjkp = 1 and kAzjkp→0 as j→ ∞.

Proof. By Theorem 5.1(a), there is a y_{1} ∈ l^{p} such that ky_{1}kp = 2 and kAy_{1}k <

1/2. IfN_{1} is large enough, thenkP_{N}_{1}y_{1}kp ≥ 1 andkAP_{N}_{1}y_{1}kp <1. Letting z_{1} :=

PN1y1/kPN1y1kp we get

z1∈l^{p}(0, N1], kz1kp= 1, kAz1kp<1.

Applying Theorem 5.1(a) to the operator AV^{N}^{1}, we see that there is an y_{2} ∈ l^{p}
such thatky_{2}kp= 2 andkAV^{N}^{1}y_{2}kp <1/4. For suﬃciently largeN_{2} > N_{1} we have
kPN2V^{N}^{1}y2kp≥1 and kAPN2V^{N}^{1}y2kp<1/2. Setting

z_{2}:=P_{N}_{2}V^{N}^{1}y_{2}/kP_{N}_{2}V^{N}^{1}y_{2}kp,
we therefore obtain

z_{2}∈l^{p}(N_{1}, N_{2}], kz_{2}kp= 1, kAz_{2}kp<1/2.

Continuing in this way we ﬁnd zj satisfying

z_{j}∈l^{p}(N_{j−1}, N_{j}], kz_{j}kp= 1, kAz_{j}kp <1/j.

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

s^{(p)}_{k} (A_{n})≥d for all n≥n_{0}. (28)
Letε >0 be any number such that

2εk^{2}< d. (29)

Choosezj as in Lemma 5.3. Obviously, there are suﬃciently largej andN such that
kP_{N}z_{l}kp≥1/2, kAP_{N}z_{l}kp< ε for l∈ {j+ 1, . . . , j+k}. (30)

Approximation Numbers of Toeplitz Matrices 15

Since PNzl ∈ l^{p}(Nl−1, l], it is clear that PNzj+1, . . . , PNzj+k are linearly indepen-
dent. Now let n ≥ N. By Theorem 5.2, there is a projection Πn of C^{n}_{p} onto
span{PNzj+1, . . . , PNzj+k}for whichkΠnkp≤k. LetIn stand for the identity oper-
ator on C^{n}_{p}. The space Im (In−Πn) = Ker Πn has the dimensionn−k and hence,
I_{n}−Π_{n}∈ Fn−k^{(n)} . Everyx∈C^{n}_{p} can be uniquely written in the form

x=γ1PNzj+1+. . .+γkPNzj+k+w with w∈Ker Πn. Thus,

kAnx−An(In−Πn)xkp=kAnΠnxkp

=kγ_{1}A_{n}(P_{N}z_{j+1}) +. . .+γ_{k}A_{n}(P_{N}z_{j+k})kp≤ |γ_{1}|ε+. . .+|γ_{k}|ε, (31)
the estimate resulting from (30). Taking into account that the sequences PNzl have
pairwise disjoint supports, we obtain from (30) that

kΠnxk^{p}p=kγ1PNzj+1+. . .+γkPNzj+kk^{p}p

=|γ_{1}|^{p}kP_{N}z_{j+1}k^{p}p+. . .+|γ_{k}|^{p}kP_{N}z_{j+k}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

kAnx−An(In−Πn)xk^{p}≤εk max

1≤m≤k|γm| ≤2εkkΠnxk^{p}≤2εk^{2}kxk^{p},

whence s^{(p)}_{k} (An) = dist (An,Fn−k^{(n)})≤ kAn−An(I−Πn)kp≤2εk^{2}.By virtue of (29),
this contradicts (28) and completes the proof.

6. Proof of Theorem 1.3.

The Hankel operator onl^{p} induced by a function a∈Mp is given by the matrix
H(a) = (aj+k−1)^{∞}_{j,k=1}.

Fora∈Mp, deﬁne ˜a∈Mp by ˜a(e^{iθ}) :=a(e^{−iθ}). Clearly,
H(˜a) = (a−j−k+1)^{∞}_{j,k=1}.
It is well known and easily seen that

T(ab) =T(a)T(b) +H(a)H(˜b) (33)
for every a, b∈M_{p}. A ﬁnite section analogue of formula (33) reads

Tn(ab) =Tn(a)Tn(b) +PnH(a)H(˜b)Pn+WnH(˜a)H(b)Wn, (34) where Pn is as in (13) andWn is deﬁned by

W_{n} :{x_{1}, x_{2}, x_{3}, . . .} 7→ {x_{n}, xn−1, . . . , x_{1},0,0, . . .}.

16 A. B¨ottcher

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

We remark that Tn(˜a) is the transposed matrix of Tn(a) and that the identity Tn(˜a) =WnTn(a)Wn holds. In particular, we have

s^{(q)}_{k} (T_{n}(a)) = minn

kT_{n}(a)−Fn−kkq:Fn−k∈ Fn−k^{(n)}

o

= minn

kT_{n}(˜a)−Gn−kkp:Gn−k ∈ Fn−k^{(n)}

o

= minn

kW_{n}(T_{n}(˜a)−Gn−k)W_{n}kp:Gn−k∈ Fn−k^{(n)}

o

= minn

kTn(a)−Hn−kkp:Hn−k∈ Fn−k^{(n)}

o

= s^{(p)}_{k} (T_{n}(a)) (35)

(note also thatW_{n} is an invertible isometry onC^{n}_{p}).

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

Lemma 6.1. IfA, B, C ∈ B(C^{n}_{p})then

s^{(p)}_{k} (ABC)≤ kAkps^{(p)}_{k} (B)kCkp for all k.

This follows easily from the deﬁnition ofs^{(p)}_{k} .

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

This is obvious from the estimates
kT_{n}^{−1}(b)Pny−T^{−1}(b)ykp

≤ kT_{n}^{−1}(b)kpkPny−Tn(b)PnT^{−1}(b)ykp+kPnT^{−1}(b)y−T^{−1}(b)ykp,
kxk^{p}≤lim inf

n→∞ kT_{n}^{−1}(b)k^{p}kT(b)xk^{p}, kξk^{q}≤lim inf

n→∞ kT_{n}^{−1}(˜b)k^{q}kT(˜b)ξk^{q}.
We now establish two propositions which easily imply Theorem 1.3.

Deﬁne χk by χk(e^{iθ}) = e^{ikθ}. Using Theorem 3.1(b) and formula (33) one can
readily see that if a∈Mp, then T(a) ∈Φ−k(l^{p}) if and only ifa =bχk and T(b) is
invertible onl^{p}.

Propostion 6.3. Ifb∈M_{p} and{T_{n}(b)}is stable onl^{p} then for every k∈Z,
lim inf

n→∞ s^{(p)}_{|k|+1}

T_{n}(bχ_{k})

>0.

Proof. We can assume thatk≥0, since otherwise we may pass to adjoints. Because
kT_{n}(χ−k)kp= 1, we obtain from Lemma 6.1 that

s^{(p)}_{k+1}

T_{n}(bχ_{k})

=s^{(p)}_{k+1}

T_{n}(bχ_{k})

kT_{n}(χ−k)kp

≥s^{(p)}_{k+1}

T_{n}(bχ_{k})T_{n}(χ−k)

=s^{(p)}_{k+1}

T_{n}(b)−P_{n}H(bχ_{k})H(χ_{k})P_{n}
,

Approximation Numbers of Toeplitz Matrices 17

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

s^{(p)}_{k+1}

Tn(b)−Fk

= infn

kTn(b)−Fk−Gn−k−1kp:Gn−k−1∈ Fn−k−1^{(n)}

o

≥ infn

kT_{n}(b)−H_{n−1}kp:H_{n−1}∈ Fn−1^{(n)}

o=s^{(p)}_{1} (T_{n}(b)).

Since {Tn(b)}is stable, we infer from (3) that lim inf

n→∞ s^{(p)}_{k+1}(Tn(bχk))≥lim inf

n→∞ s^{(p)}_{1} (Tn(b))>0.

Proposition 6.4. If b∈M_{p} and{T_{n}(b)}is stable onl^{p} then for everyk∈Z,

n→∞lim s^{(p)}_{|k|}(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^{(p)}_{k} (T_{n}(bχ_{k})) = s^{(p)}_{k}

T_{n}(χ_{k})T_{n}(b) +P_{n}H(χ_{k})H(˜b)P_{n}

≤ kTn(b)kps^{(p)}_{k}

Tn(χk) +PnH(χk)H(˜b)PnT_{n}^{−1}(b)
.
Put An:=Tn(χk) +PnH(χk)H(˜b)PnT_{n}^{−1}(b). We have

A_{n} =

∗ C_{n}
In−k 0

=

∗ 0 In−k 0

+

0 C_{n}

0 0

=:B_{n}+D_{n},

the blocks being of sizek×(n−k), k×k, (n−k)×(n−k), (n−k)×k, respectively.

Clearly,Bn has rankn−kand thusBn∈ Fn−k^{(n)}. It follows that
s^{(p)}_{k} (A_{n}) =s^{(p)}_{k} (A_{n}−B_{n}) =s^{(p)}_{k} (D_{n})≤ kD_{n}kp=kC_{n}kp,
and we are left with showing thatkCnkp→0.

Letbn(n∈Z) be the Fourier coeﬃcients ofb, letej ∈l^{p} be the sequence whose
only nonzero entry is a unit at thejth position, and recall the notation (23). We have
C_{n} = (c^{(n)}_{jl} )^{k}_{j,l=1}, and it is easily seen thatc^{(n)}_{jl} equals (b_{−k+j−1}, . . . , b_{−k+j−n}) times
the (n−k+l)th column ofT_{n}^{−1}(b):

c^{(n)}_{jl} = (b−k+j−1. . . b−k+j−n)T_{n}^{−1}(b)P_{n}en−k+l=

P_{n}f_{jk}, T_{n}^{−1}(b)P_{n}en−k+l

where

fjk:=n

b−k+j−1, b−k+j−2, b−k+j−3, . . .o

=T(χ−k+j−1)T(˜b)e1∈l^{q}.
Consequently,

c^{(n)}_{jl} =

T_{n}^{−1}(˜b)P_{n}f_{jk}, en−k+l

=

T^{−1}(˜b)f_{jk}, en−k+l

+

T_{n}^{−1}(˜b)P_{n}f_{jk}−T^{−1}(˜b)f_{jk}, en−k+l

. (36)

18 A. B¨ottcher

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

kT_{n}^{−1}(˜b)Pnfjk−T^{−1}(˜b)fjkkq (37)
(note thatken−k+lkp = 1). Our assumptions imply that{T_{n}(˜b)}is stable onl^{q}. We
so deduce from Lemma 6.2 that (37) tends to zero as n→ ∞.

Thus, each entry of thek×kmatrixCn approaches zero asn→ ∞. This implies that kCnkp→0.

Now letabe as in Theorem 1.3. SinceT(a)∈Φ−k(l^{r}) for allr∈[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 onl^{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^{0}_{p}denote the collection of all symbolsb∈M_{p}for which{T_{n}(b)}is stable
on l^{p} and let Π_{p} be the set of all symbolsa ∈ M_{p} such that aχ_{−k} ∈ Π^{0}_{p} for some
k∈Z. Notice that

Π_{p}= Π_{q}⊂ [

r∈[p,q]

Π_{r}
and

G(C+H^{∞})∪G(C+H^{∞})∪G(P QC)⊂Π26=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∈Πp andT(a)∈Φk(l^{p}) then
Σ^{(p)}(Tn(a)) :=n

s^{(p)}_{1} (Tn(a)), . . . , s^{(p)}_{n} (Tn(a))o
has the |k|-splitting property.

We also note that the proof of Proposition 6.4 gives estimates for the speed of
convergence ofs^{(p)}_{|k|}(T_{n}(bχ_{k})) to zero. For example, ifP

n∈Z|n|^{µ}|b_{n}| <∞ (µ >0),
then the ﬁnite section method is applicable toT(b) on the spacel^{2,µ} of all sequences
x={x_{n}}^{∞}n=1 such that

kxk2,µ:=

X^{∞}

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
ken−k−lk^{2,−µ}= (n−k+l)^{−µ}=O(n^{−µ}),

the proof of Proposition 6.4 implies the following result.

Corollary 6.6. If P

n∈Z|n|^{µ}|an|<∞for someµ >0andT(a)∈Φk(l^{p})then
s^{(p)}_{|k|}(T_{n}(a)) =O(n^{−µ}) as n→ ∞.