vol5_pp104-125. 325KB Jun 04 2011 12:06:14 AM

(1)

The Electronic Journal of Linear Algebra.

A publication of the International Linear Algebra Society. Volume 5, pp. 104-125, December 1999.

ISSN 1081-3810. http://math.technion.ac.il/iic/ela

ELA

TOEPLITZ BAND MATRICES WITH EXPONENTIALLY GROWING

CONDITION NUMBERS

A. BOTTCHERy

AND S. GRUDSKY z

Abstract. The paper deals with the spectral condition numbers (Tn(b)) of sequences of

Toeplitz matricesTn(b) = (b j,k)

n j;k =1 as

ngoes to innity. The function b(e i) =

P k

b k

e ik is

referred to as the symbol of the sequence fT n(

b)g. It is well known that (T n(

b)) may increase

exponentially if the symbolbhas very strong zeros on the unit circleT=fz 2C :jzj= 1g, for

example, ifbvanishes on some subarc ofT. Ifbis a trigonometric polynomial, in which case the

matricesTn(b) are band matrices, thenbcannot have strong zeros unless it vanishes identically. It

is shown that the condition numbers(T n(

b)) may nevertheless grow exponentially or even faster to

innity. In particular, it is proved that this always happens ifbis a trigonometric polynomial which

has no zeros onTbut nonzero winding number about the origin.

The techniques employed in this paper are also applicable to Toeplitz matrices generated by rational symbolsband to the condition numbers associated withl

pnorms (1

p1) instead of

thel 2norm.

Keywords.Toeplitz matrix, band matrix, matrix norm, condition number

AMSsubject classications.15A12, 15A60, 47B35, 65F35

1. Introduction and main results.

Let l p n (1

p1) be the linear space C

n with the l

pnorm,

kxk p=

X

j jx

j j

p

1=p

for 1p<1; kxk

1= max j

jx j

j:

Every nn matrix A

n induces an operator on l

p

n, and we let kA

n k

p stand for the

norm of this operator,

kA n

k p= sup

x6=0 kA

n xk

p =kxk

p :

The condition number p(

A

n) is dened by

p(

A n) =

kA n

k p

kA ,1 n

k p

;

where, by convention,kA ,1 n

k p=

1ifA

n is not invertible.

Leta be a rational function without poles on the complex unit circle T=fz2 C :jzj= 1g. We denote byfa

k g

k 2Zthe sequence of the Fourier coecients of a, that

Received by the editors on 14 October 1999. Accepted for publication on 29 November 1999.

Handling Editor:Ludwig Elsner. This work was supported by DFG-Kooperationsprojekt 436 RUS 113/426 for German and Russian scientists within the \Memorandum of Understanding" between DFG and RFFI.

yFaculty of Mathematics, Technische Universitat Chemnitz, D - 09107 Chemnitz, Germany

(aboettch@mathematik.tu-chemnitz.de).

zFaculty of Mechanics and Mathematics, Rostov-on-Don State University, Bolshaya Sadovaya

105, 344 711 Rostov-on-Don, Russia (grudsky@aaanet.ru), partially supported by RFFI Grant 98-01-01023.

(2)

ELA

Condition Numbers of Toeplitz Band Matrices 105

is,

an= 12

Z 2

0

a(ei)e ,in

d= 1

2i Z

T a(z)z

,n,1 dz:

(1)

Thenn Toeplitz matrix with the symbola is the matrixTn(a) = (aj ,k)

n,1

j;k=0. If 2C, thenTn(a),I=Tn(a,). It is well known that, for xeda, the properties

ofTn(a,) depend very sensitively on2C. This paper is devoted to the behavior

of the condition numbers (Tn(a,)) as n goes to innity. Notice thatTn(a) is a

band matrix if and only ifais of the forma(z) = Pm

k=,m akzk.

Given annnmatrixAn, for instance,An =Tn(a,), we let 1(

An)::: n(An) denote the singular values ofAn, that is, the eigenvalues of (A

nAn)

1=2. The lp

analogues of the singular values are the approximation numbers. Forj2f0;1;:::;ng,

we denote by F (n)

j the nn matrices of rank at most j. Then the approximation

numbers

(p) 1 (

An)::: (p)

n (An)

are dened by

(p)

j (An) = distp(An;F (n)

n,j) := inf

fkAn,Fnkp:Fn2F (n)

n,j g:

It is well known that (2)

j (An) =j(An). Moreover, we have

(p) 1 (

An) =kA ,1

n k

,1

p ;

(p)

n (An) =kAnkp:

Consequently,

p(An) =

(p)

n (An)

(p) 1 (

An)

= kAnkp

(p) 1 (

An) :

(2)

In the case whereAn =Tn(a,), the normskAnkp have a nite limit, the pnorm

of the innite Toeplitz matrixT(a,) = (aj ,k

,j;k) 1

j;k=0. Thus, the asymptotic

behavior of the condition numbers(Tn(a,)) is entirely governed by the smallest

approximation number (p) 1 (

Tn(a,)).

We orient the unit circle T counter-clockwise. Then the image a(T) is a

natu-rally oriented closed curve in the complex plane C. For 2 C na(T), we denote

by wind(a;) the winding number of the curve a(T) about. Clearly, the integer

wind(a;) is constant on each connected component ofC na(T).

A classical result by Gohberg and Feldman [12] says that liminf_n

!1

(p) 1 (

Tn(a,))>0

or, equivalently,

limsup_n

!1

(3)

ELA

106 A.BottcherandS.Grudsky

if and only if2C na(T) and wind(a;) = 0. Only recently, Roch and Silbermann

[18] discovered that if2C na(T) and wind(a;) =k6= 0, then jkjsingular values

of T n(

a,) go to zero, while the remaining n,jkj singular values stay away from

zero (this is the so-called \splitting phenomenon"). In [4], this result was proved by completely dierent methods and extended to approximation numbers. Thus, for

2C na(T) and wind(a;) =k, we have

lim

n!1

(p) jk j(

T n(

a,)) = 0; liminf n!1

(p) jk j+1(

T n(

a,))>0:

We here sharpen the last result as follows.

Theorem 1.1. If 2C na(T)andwind(a;) =k6= 0,thenthereareconstants C2(0;1)and2(0;1)independent ofn suchthat

(p) jk j(

T n(

a,))Ce ,n

for all n1:

This theorem in conjunction with (2) implies that in the case wind(a;)6= 0 the

condition numbers increase at least exponentially:

Corollary 1.2. If2C na(T)andwind(a;)6= 0,thenthere existconstants D2(0;1)and2(0;1)independent ofn suchthat

p(

T n(

a,))De n

for all n1:

It turns out that the constantcan be determined. Let 0<r

<1<R

<1

be any two numbers such that a(z), has no zeros and no poles in the annulus fz2C :r

jzjR

g. We will show that Theorem 1.1 and Corollary 1.2 are true

with

= min(log 1 r

;logR

) :

(3)

Of course, the constantsC andD also depend on.

Corollary 1.2 provides us with a lower estimate for the condition numbers and raises the question about upper estimates. Let, for example,a(z) =z. Thena(T) =T

and wind(a;0) = 1. Obviously, T n(

a,0) = T n(

a) is not invertible and therefore

p( T

n(

a)) =1 for all n1. This shows that nontrivial universal upper estimates

do not exist.

The functiona,belongs toL 2(

T) and, by Parseval's equality,

ja 0

,j 2+

X

j6=0 ja

j j

2

= 12 Z

2

0 ja(e

i) ,j

2

d= 12

ka,k 2 2 :

(4)

The following is some kind of an upper estimate.

Theorem 1.3. If 2C na(T)andwind(a;)6= 0,then

kT ,1 n (

a,)k p

n jdetT

n( a,)j

1

p

2

ka,k 2

n,1

:

(4)

ELA

Sincen<e

n=10 for all suciently large

n, we can put

= 110 +log

1

p

2

ka,k 2

and, fornlarge enough, rewrite (5) in the form

p(

T n(

a,))

1

jdetT n(

a,)j Ee

n

with some constant E 2 (0;1) independent of n. Notice that formulas for the

determinants of rationally generated Toeplitz matrices are available (see, e.g., [3], [8], [9], [10]).

The following example illustrates Corollary 1.2 and Theorem 1.3. Let a(z) = z,4z

,1 and

= 0. Since a(z) =z ,1(

z,2)(z+ 2), we see that wind(a;0) = ,1.

Elementary computations give detT

n( a) =

2n if

n is even,

0 if n is odd.

From (4) we getkak 2

= p

2= p

17, and Theorem 1.3 so provides us with the estimate

kT ,1 n (

a)k p

1

p

17n p

17 2

! n

<

1 4ne

0:724n <

1 4e

0:73n

for all suciently large evenn. On the other hand, Corollary 1.2 and (3) show that

kT ,1 n (

a)k p

De

n(log2,0:001) >De

0:69n

with some constantD>0 for alln. In summary, there areD 1

;D 2

2(0;1) such that

D 1

e 0:69n

p( T

n(

a))D 2

e

0:73n for all even n;

whereas

p(

T n(

a)) =1 for all odd n:

Let spT n(

a) stand for the spectrum ( = set of eigenvalues) ofT n(

a). The limiting

set

(a) = lim sup n!1

spT n(

a)

is the set of all partial limits of the sequencefspT n(

a)g 1

n=1. In other words,

2(a)

if and only if there are n 1

<n 2

<n 3

<::: and n

k 2spT

n k(

a) such that n

k !.

The set (a) is known from the work of Schmidt and Spitzer [20], Hirschman [14],

and Day [11]. It can be described in terms of only the functiona, and one can show

(5)

ELA

most their endpoints in common. Thus, (a) is always a \thin" set. Ifa(z) =z and

a(z) =z,4z

,1 are as in the above examples, then (a) =

f0gand (a) = [,4i;4i],

respectively.

Theorem 1.4. If 2 C n(a(T)[(a)) and wind (a;) 6= 0, then there are

constants D1;D2

2(0;1) and;2(0;1) independent of nsuch that

D1e n

p(Tn(a

,))D 2e

n

for all suciently large n.

As the following result shows, 2(Tn(a

,)) may grow arbitrarily fast on (a).

We letN denote the natural numbers.

Theorem 1.5. Leta(z) =z+z,1=4. Thena(

T) is an ellipse with the foci,1 and

1, and (a) = [,1;1]. Given any function ':N !N, for example, '(n) = exp(n n),

there exists a point 2(a) such that p(Tn(a

,))<1for all n1 but

p(Tnk(a

,))> '(n

k) for innitely many nk:

The results quoted above provide an idea of the behavior ofp(Tn(a

,)) forin

the connected components ofC na(T) whose winding number is nonzero. As already

said, Gohberg and Feldman showed thatp(Tn(a

,)) remains bounded on the other

connected components ofC na(T). In [6], we proved that if wind(a;) = 0, then the

condition numbersp(Tn(a

,)) even converge to a nite limit asn!1.

The behavior of p(Tn(a

,)) for 2 a(T) is more intricate. Much work on

this problem has been done in the case whereais real-valued. Let us assume thata

is a real-valued nonconstant continuous function onT. Then the matricesT

n(a) are

Hermitian and (a) can be shown to be the line segment a(T) = [mina;maxa] =:

[m;M]. Ifdoes not belong to this segment, then it is easy to prove that2(Tn(a ,))

remains bounded asn! 1(Brown-Halmos theorem). If is one of the endpoints

morM of the segment [m;M], then a(z),has zeros on Tand all these zeros are

of even order. Suppose all these zeros have nite orders and let 2be the maximal order. The outgrowth of the works [13], [16], [17], [21], [23] is that there are constants

D1;D2

2(0;1) such that

D1n 2

2(Tn(a

,))D 2n

2:

(6)

Rosenblatt [19] was probably the rst to understand that if2fm;Mganda(z),

vanishes identically on some subarc ofT, then 2(Tn(a

,)) increases exponentially.

Serra [22] observed that if2fm;Mgthen always 2(Tn(a

,)) =O(e

n) for some

constant2(0;1) asn!1.

Some general results on the condition numbers of non-Hermitian Toeplitz matrices are in [5]. There we proved, in particular, that if ais rational and 2 a(T), then

necessarily 2(Tn(a

,)) Dn with some constant D 2 (0;1) independent of n.

Moreover, ifa(z),has a zero of order2N, then even 2(Tn(a

,))Dn with

(6)

ELA

Finally, suppose ais both rational and real-valued. Then again (a) =a(T) =

[m;M]. Rational functions cannot vanish on an open arc unless they vanish identi-cally. Thus, we cannot expect that2(Tn(a

,)) grows exponentially for2fm;Mg.

From (6) we infer that indeed2(Tn(a

,)) increases at most polynomially ifis one

of the endpointsmandM. As the following result shows, things change dramatically for pointsin the interior of [m;M].

Theorem 1.6. Leta(z) =z+z,1. Then a(

T) = (a) = [,2;2], and given any

function ':N !N, there exist an2(,2;2) such that

p(Tn(a

,))<1 for all n1

but

p(Tn k(a

,))> '(n

k) for innitely many nk:

We will establish several new results on the condition numbers2(Tn(a

,)) for

2a(T).

Suppose thata(t),has only one zero onT. We label each connected component

ofC na(T) with the winding number ofa(T) about the points of the component. Our

assumption that a(t), has only a single zero amounts to the requirement that

is a point ona(T) and thata(T) passes through exactly once. In particular,a(T)

has no self-intersection at . In a neighborhood of , the curve a(T) is an oriented

analytic arc. LetM + andM

,

be the connected components of

C na(T) which lie

on the left and the right of, respectively.

In what follows we write xn y

n if yn

6

= 0 for all suciently large n and lim(xn=yn) = 1, whilexn

'y

n means thatyn

6

= 0 and 1=Cx n=yn

C with some

constantC2(0;1) for all nlarge enough.

Theorem 1.7. Letabe a rational function without poles onTand let 2a(T).

Suppose b(t) := a(t), has exactly one zero t 0

2 T and let be the order of this

zero. Furthermore, let mdenote the winding number ofM+ . Then

2(Tn(a

,))'n if

,

2

m

+ 1 2

;

(7)

and there are constantsC >0 and >0 such that

2(Tn(a

,))Ce

n if m < ,

2

or m >

+ 1 2

:

(8)

We denote byS(a) the points ofa(T) at whicha(T) has self-intersections.

Equiv-alently,2S(a) if and only ifa(t),has at least two dierent zeros onT. In many

casesS(a) is a nite subset ofa(T), but as functions of the form a(t) =d(B(t)) with

a rational functiond and a nite Blaschke productB show, S(a) may also coincide with all of a(T). If 2a(T)nS(a), thena(t),has a single zero onT. The order

(7)

ELA

The spectrum of the Toeplitz operatorT(a) given onl

2 by the innite Toeplitz

matrix (a j,k)

1 j;k =0 is

spT(a) =a(T)[f2C na(T) : wind(a;)6= 0g:

Let@spT(a) stand for the boundary of spT(a).

Theorem 1.8. If 2a(T)nS(a) islocatedon@spT(a),then

2(

T n(

a,)'n ()

:

Theorem 1.9. Suppose() = 1 forall2a(T)nS(a). Then

2(

T n(

a,))'n

if 2a(T)nS(a)lies on@spT(a),whereas

2(

T n(

a,)) inceasesatleast exponentially

if 2a(T)nS(a)isnotlocatedon@spT(a).

The rest of the paper contains the proofs of the theorems stated above. The two main ingredients to the proof of Theorem 1.1 are the so-called nite section method for Toeplitz operators on certain spaces with exponential weights on the one hand and the approach to the approximation numbers developed in [4] on the other. Corollary 1.2 is an immediate consequence of Theorem 1.1. Theorem 1.3 is nothing but a simple application of Hadamard's inequality. The proof of Theorem 1.4 is based on Theorem 1.3 and on Day's formulas for the determinants ofT

n(

a,) and the limiting set (a).

To prove Theorems 1.5 and 1.6, we rst make use of the fact that the eigenvalues of tridiagonal Toeplitz matrices are explicitly available, and we then nd the desired

by employing a standard construction of number theory. Theorem 1.7 will be derived from some rather subtle results of [8]. Theorems 1.8 and 1.9 are simple consequences of Theorem 1.7. In the last section of the paper we present a few additional examples and results on the condition numbers of Toeplitz matrices whose symbols have zeros. The paper has an appendix. This appendix does not contain terribly new results, but it tells a nice story we have not yet seen in this form in the literature.

2. Spaces with exponential weight. In this section we prepare the proof of

Theorem 1.1. The results of this section are known to specialists. As we do not know an explicit reference, we outline the proofs for the reader's convenience.

Given a real number , we denote by l

p; the Banach space of all sequences x=fx

n g

1

n=0for which

kxk p p;=

1 X

n=0 e

pn jx

n j

p

<1 (1p<1);

kxk

1;= sup e

n jx

n

(8)

ELA

Forn2N, letP n and

Q

n be the projections P

p;will be denoted by l

p;

n . We identify l

p;

n and the space C

n with the l

p; norm. Given Banach spaces

X andY, we letB(X;Y) stand for the

Banach space of all bounded linear operators ofX to Y, and we abbreviateB(X;X)

toB(X).

For a rational function awithout poles on T, we denote byT(a) andH(a) the

innite Toeplitz and Hankel matrices generated bya:

T(a) = (a

induce bounded operators onl p;.

<1, and it can be veried straightforwardly that kT(

n) k

p; = O(e

jnjjj). This shows that

T(a) 2 B(l

p;). The proof for

H(a) is

analogous.

Proposition2.2. Letjjand 1p1. Ifwind(a;0) = 0, thenT(a) is

invertible onl p;.

Proof. On writingaas the quotient of two polnomials, it is easily seen thatacan

be represented in the form

a(t) =a

and poles in fz 2 C : jzj 1g. The representation (11) is called a Wiener-Hopf

factorization. The Fourier coecients ofa ,

+ admit estimates analogous

to (9), and therefore we obtain as in the proof of Proposition 2.1 thatT(a ,),

(9)

ELA

Lemma 2.3. If >0and1p1, then

Proof. Straightforward.

Theorem 2.4. Letjjand1p1. If wind(a;0) = 0,then

the theorem for 0. In the case = 0, the theorem is Gohberg and Feldman's

[12]. The= 0 result implies in particular thatT n(

a) is invertible for all suciently

largen. Now letx2l

By Proposition 2.2, the second term on the right of (15) does not exceedM 2

kxk p;

with some constantM 2

<1independent of n. Taking into account (12) we obtain

the estimate

for the rst term on the right of (15), and since

kT

for all suciently largen, it follows that (16) is not larger than

Ce

Propositions 2.1 and 2.2 in conjunction with (13) imply that (17) can be estimated from above byM

1 kxk

p; with some M

1

<1independent ofn. In summary,

kT

which completes the proof.

3. Exponential decay of approximation numbers.

The purpose of this sec-tion is to prove Theorem 1.1.

Thus, let2C na(T) and suppose wind(a;) =k6= 0. For the sake of

(10)

ELA

to adjoints. Putb=a,, chooser=r

this can be veried directly by computing thejkentries of each side or can be derived

from a more general result on the multiplication of nite Toeplitz matrices (see, e.g., [9, Proposition 2.12]). Hence,

Ifnis large enough, thenT n(

c) is invertible and

kT

due to Theorem 2.4. Since distp( AB;F

p;, boundedly into itself. Since H(

k) has only

nitely many nonzero entries, we arrive at the conclusion that the rst factor of (21) is nite. The middle factor of (21) remains bounded by virtue of (18). From (13) we nally see that the third factor of (21) isO(e

(11)

ELA

4. Upper estimates.

This section contains the proof of Theorem 1.3. Let

b = a,, r;R ; be as in Section 3. We abbreviate detT n(

b) to D n(

b). What we

must show is that

kT

There is nothing to prove in caseD n(

j+k times the determinant of the

matrix arising fromT n(

b) by deleting thekth row and thejth column. By Hadamard's

inequality (see, e.g., [2, p. 3]) and by (4),

panalogue of the Frobenius norm we obtain, with obvious

modications forp= 1 and p=1,

This completes the proof of Theorem 1.3.

5. Outside the limiting set.

The proof of Theorem 1.4 is based on two results by Day [10], [11].

Suppose rst thatT n(

a) is a triangular matrix. Then (a) =fa 0

g. So let6=a 0.

It follows thatD n(

with certainD 2

2(0;1) and2(0;1) independent ofn.

So assumeT n(

(12)

ELA

Condition Numbers of Toeplitz Band Matrices 115 Theorem 5.1. (Day [10]) Let a be as above and suppose the zeros %1;:::;%s

are pairwise distinct. Ifs < h, then Dn(a) = 0. Ifsh, then

Dn(a) =X

M CM!nM;

where the sum is over all,s

h

subsetsM f1;:::;sgof cardinalityjMj=hand, with

M=f1;:::;sgnM; G=f1;:::;gg; H =f1;:::;hg;

the constants are given by

!M = (,1)s ,h

Y

j2M

%j

CM = Y j2M l2H

(%j,l) Y

i2G k 2M

(i,%k) Y

i2G l2H

(i,l) ,1

Y

j2M k 2M

(%j,%k) ,1:

In order to determine the limiting set (a), we have to consider

a(z),=Fm(z;)

g

Y

j=1

1,

z j

,1 h

Y

j=1

(z,j) ,1

with

Fm(z;) :=Ys

j=1

(z,%j),

g

Y

j=1

1,

z j

h Y

j=1

(z,j):

If 6= 0 and 6= Qg

j=1(

,j), then the polynomial Fm(;) has the degree m :=

max(s;g+h).

Theorem 5.2. (Day [11]) Let a be the function introduced above. For 6= 0

and6= Qg

j=1(

,j), label the zeros z

1();:::;zm() ofFm(z;) so that jz

1() jjz

2()

j:::jzm()j:

Then

(a) = clos

8 <

:

2C nf0;

g

Y

j=1

(,j)g:jzg()j=jzg +1()

j 9 =

;

:

We now proceed to the proof of Theorem 1.4. LetE(a) denote the set of all2C

for which Fm(;) has multiple zeros. If2E(a), thena(z) =and a

0(z) = 0 for

somez2C. Asa

0 has only nitely many zeros in

C, it follows that E(a) must be a

nite set.

Suppose 2 C n(a(T)[(a)[E(a)) and wind(a;) 6= 0. Asssume also that

6= 0 and 6= Qg

j=1(

,j). Then, by Theorem 5.2, jzg()j < jzg +1()

(13)

ELA

the zeros z 1(

);:::;zm() of a, are paiwise distinct for 2= E(a), we can have

recourse to Theorem 5.1 to compute Dn(a,). The role of the numbers % 1

;:::;%s

of Theorem 5.1 is now played by the zerosz 1(

);:::;zm(). Put M 0=

f1;2;:::;hg.

Sincejzg()j<jzg +1(

)j, it follows that

j!M 0

j=jzg +1(

)j:::jzm()j

is strictly larger than all otherj!Mj's. Consequently,

jDn(a,)j=jCM 0

jj!M 0

jn(1 +O(n))

with some constant2(0;1). This implies that

1

jDn(a,)j

D()e ()n

(23)

with certain constantsD()2(0;1) and()2(0;1) independent ofn. Combining

(23) and Theorem 1.3 we arrive at the upper estimate of Theorem 1.4. We are left with the case where

0

2C n(a(T)[(a)), wind(a; 0)

6

= 0, but

0

2E(a)[f0;

g

Y

j=1

(,j)g:

SinceE(a) is nite, there is a small circle , =f2C :j, 0

j="gsuch that

2=a(T)[(a)[E(a)[f0;

g

Y

j=1

(,j)g

for all2,. For the points on the circle , we have the estimate (23). The zeros z

1(

);:::;zm() are continuous functions of 2 ,. From the explicit form of the

constants in Theorem 5.1 we therefore see that the constantsD() and () in (23)

can also be chosen to be continuous functions of. Thus, since , is compact, there

are constantsD2(0;1) and2(0;1) such that

1

jDn(a,)j

Den

(24)

for all2,. Ifnis suciently large, then, by the denition of (a),Dn(a,)6= 0

for all2C satisfyingj, 0

j". Hence 1=Dn(a,) is an analytic function of

on the disk :=f2C :j, 0

j"g. From the maximum modulus principle we

therefore deduce that (24) holds for all2 and, in particular, for=

0. This in

conjunction with Theorem 1.3 gives the upper estimate in Theorem 1.4.

(14)

ELA

6. Inside the limiting set.

This section is devoted to the proofs of Theorems 1.5 and 1.6.

Thus, let rst a(z) =z+z ,1

=4. Because

a(e

i) = (1 + 1

=4)cos+i(1,1=4)sin;

it is clear that a(T) is an ellipse with the foci ,1 and 1. The set (a) can be

determined using Theorem 5.2, but since T n(

a) is tridiagonal, we can also nd (a)

from computing the eigenvalues of T n(

a). Namely, it is well known (see, e.g., [13])

that if

a(z) =a ,1

z ,1+

a 0+

a 1

z;

then the eigenvalues ofT n(

a) are

j(

T n(

a)) =a 0+ 2

p a

,1 a

1cos j n+ 1 (

j= 1;:::;n):

Consequently, in our special case we have the eigenvalues

j(

T n(

a)) = cos j n+ 1 (

j = 1;:::;n);

(25)

which implies that (a) = [,1;1].

Now let 2 (,1;1). There is a y 2 (0;) such that = cosy. The matrix T

n(

a,) has the eigenvalues

j(

T n(

a,)) = cos j n+ 1

,cosy (j= 1;:::;n):

(26) Obviously,

j j(

T n(

a,))j= 2 sin 12

j n+ 1

,y

sin 12

j n+ 1 +

y

, j n+ 1

:

(27) Let

(n) := min 1jn

j j(

T n(

a,))j:

(28)

We denote by%() the spectral radius. Since

kT ,1 n (

a,)k p

%(T ,1 n (

a,)) = 1=(n);

it suces to prove that there is ay2(0;) such that(n)>0 for alln1 and

(n k)

<1='(n k)

(15)

ELA

To nd the desiredy, we use a standard construction (see, e.g., [15, p. 6]). Pick

any natural number N 1

1 and choose natural numbers N 1

successively by requiring that

'

Obviously, the number y= is irrational. This implies that none of the eigenvalues

(26) is zero, whence(n)>0 for alln1. Since

it follows that

10,N1+ 10,N1,N2+

which shows that 0<

the last inequality resulting from (29). Thus, by (27),

(n)j

This completes the proof of Theorem 1.5

The proof of Theorem 1.6 is the same as the one of Theorem 1.5. Indeed, if

a(z) =z+z

,1, then the eigenvalues of T

n(

a) are (25), and hence we can repeat the

above reasoning starting with (26).

7. Inside the essential spectrum.

In this section we prove Theorems 1.7 to 1.9. The following result is Theorem 1.7 in analytic language.

Theorem 7.1. Letabe a rational function without poles onTand let 2a(T).

winding number about the origin is zero. Then

(16)

ELA

and there are constantsC >0 and >0 such that

2(Tn(a

,))Ce

n if k <

, or k >0:

(32)

Proof. Theorem 4.1(a) of [5] implies that2(Tn(b)) Cn

in either case.

Suppose, k0. We then can write

b(z) = (t,t 0)

,jk j

1,

t0

t

jk j

c(t)

with,jkj0 andjkj0, and from Theorem 7.87(a) of [8] we deduce that

2(Tn(b)) =O(n

,jk j+jk j) =O(n):

At this point the proof of (31) is complete. Now supposek <, or k >0. Since

eb(t) :=b(1=t) = (1 ,t

0t)

t,,kc(1=t); t 2T;

and 2(Tn(b)) = 2(Tn(

eb)), it suces to consider the case k > 0. Put(t) = (t ,

t0)

tk. Because the Fourier coecients

n ofvanish forn

k,1, we obtain as in

Section 3 that

k(Tn(b)) kck

W kA

nQn,k k

2;

where

An =Tn() +PnH()H( e

c)PnT ,1 n (c):

AsTn()Qn,k = 0 and asH() has only nitely many nonzero entries, we can also

proceed as in Section 3 to conclude that

k(Tn(b)) =O(e

,n) as n !1;

which gives (32).

We are now going to translate Theorem 7.1 into the language of Theorem 1.7.

Lemma 7.2. If m is the winding number of M+

, then M ,

has the winding

numberm,1.

Proof. Let n be the winding number of M,

. Fix a point 2 M

+

suciently

close to. In a small neighborhood of, changea(T) to a curvewhich rst follows

, then goes intoM +

and encircles like a half-circle, and which nally goes back

to and again follows. Clearly,

wind(;) =m,1:

On the other hand, sinceandM,

are contained in the same connected component

ofC n, we have

(17)

ELA

Consequently,n=m,1.

Now letr2(0;1) and considerb(rt) =a(rt),fort2T.

Lemma 7.3. Letr 2(0;1) be suciently close to 1 and suppose a point moves

along the curvea(rT). Then, in a small neighborhood of, this point is rst inM + ,

then it encirclesexactly[(,1)=2] times in the clockwise direction, after which the

point is again in M+ .

Proof. We havea(t0) =and

a(z) =+ 1_!a()(t 0)(z

,t 0)

+O((z ,t

0) +1)

with an analytic function O((z ,t 0)

+1) as z ! t

0. Thus, locally a acts as the

function

z7!+ 1

!a()(t 0)(z

,t 0)

:

(33)

LetU C andV C be suciently small open neighborhoods oft

0 and 0,

respec-tively. Clearly, the images of U\T andU \rTunder the map (33) have the same

structure as the images ofV \C andV \rC under the mapz 7!z

, whereC is a

large circle in the upper half-plane that touches the real line at the origin. It can be checked straightforwardly that in the latter case the situation is as described in the lemma.

Proof of Theorem1.7. Write bin the form (30). We then have

b(rt) = (rt,t 0)

rktkc(rt); t 2T:

Letr2(0;1) be suciently close to 1. Then =2a(rT) and

wind(a(rt);) = wind (b(rt);0) (34)

= wind((rt,t 0)

;0) +k+ wind(c(rt);0) = 0 +k+ 0 =k:

Evidently, maxt2T

ja(rt),a(t)j!0 asr!1,0. This in conjunction with Lemma

7.3 shows that if2M ,

is suciently close to, then

wind(a(rt);) = wind(a(rt);) = wind (a(t);),

,1

2

:

(35)

From (34) and (35) we obtain

k= wind(a(t);),

,1

2

:

By Lemma 7.2, wind (a(t);) =m,1. Consequently,

k=m,1,

,1

2

(18)

ELA

Since

,m,1,

,1

2

0 () ,

2

m

+ 1 2

;

(7) and (8) follow from (31) and (32). This completes the proof of Theorem 1.7. Proof of Theorem 1.8. If2a(T)nS(a) belongs to @spT(a), thenM

+ orM

,

has the winding number zero. Thus, by Lemma 7.2, M,

has the winding number ,1 or 0. Since [(,1)=2]0 and,[(+ 2)=2],1 for every 2N, the assertion

follows from Theorem 1.7. The proof of Theorem 1.8 is complete.

Proof of Theorem1.9. The rst part of the assertion is immediate from Theorem 1.8. So suppose 2 a(T)n(S(a)[@spT(a)) and letm be the winding number of

M,

. By Lemma 7.2, the winding number ofM +

ism+ 1. Asm

6

= 0 andm+ 16= 0,

we havem1 orm,2. Theorem 1.7 with= 1 completes the proof of Theorem

1.9.

Example 7.4. Leta(t) = (t+ 1)3. We have

a(T) =A 1

[f,1g[A 2

[f0g;

where

A1= fa(e

i) :

,2=3< <2=3g;

A2= fa(e

i) : 2=3< < or

, < <,2=3g:

Clearly,S(a) =f,1g,() = 1 for2A 1

[A

2, and() = 3 for = 0. Theorem

1.7 implies that

2(Tn(a

,))'

n for 2A 1;

n3 for = 0;

and that2(Tn(a

,)) increases at least exponentially for2A 2.

Theorems 7.1 and 1.7 are not applicable to the case=,1. We have

a(t) + 1 = (t+ 1)3+ 1 = (t+ 2)(1

,t!)(1,t! 2)

with!=,1=2 +i p

3=2. Put=!and =!2. It is easily seen that

T,1

n (a+ 1) =T

n(c)Tn(')

where

c(t_{) = 12} 1 X

k =0

,

t

2

k

; '(t) = 1 X

k =0

k +1 ,

k +1

, t k:

Clearly,kT n(c)

k 2

'1. Since

kT n(')

k 1=

kT n(')

k 1=

n,1 X

k +1 ,

k +1

,

(19)

ELA

it follows that

kT n(')

k 2

kT n(')

k 1=2 1

kT n(')

k 1=2

1 =O(n):

Considering the rst column of ofTn('), we see that actually kT

n(') k

2 'n.

Example 7.5. There areasuch that2(Tn(a

,)) does not grow at most

poly-nomially although2@spT(a). Trivial examples are given by real-valued functions

a. Here is a less trivial example. Let

a(t) =t2 ,t

,2

,i(t,t ,1):

Then

a(ei) =e2i ,e

,2i ,i(e

i ,e

,i) = 2sin+ 2isin2:

Obviously, 02@spT(a)\S(a). Since T

n(a) is skew-symmetric and thusDn(a) = 0

ifnis odd, we see that2(Tn(a

,0)) =

2(Tn(a)) = +

1ifnis odd.

We nally consider the simplest case of points inS(a)\@spT(a): we assume that

b(t) =a(t),has exactly two simple and distinct zeros onTand that, in a suciently

small neighborhood ofand after appropriate translation, rotation, and straightening,

a(T) looks like the union of the positively orientedx-axis and the positively oriented

y-axis in a neighborhood of the origin of thex;y-plane. The winding numbers of the four connected components ofC na(T) which touch associate four integersm

++,

m,+, m,,, m+, with the quadrants of thex;y-plane. Since

2@spT(a), at least

one of the four numbersm; must be zero. Taking into account Lemma 7.2, we see

that there are exactly three cases:

case 1: m++= 0; m,+= 1; m,,= 0; m+,= ,1;

case 2: m++= ,1; m

,+= 0; m,,= ,1; m

+,= ,2;

case 3: m++= 1; m,+= 2; m,,= 1; m+,= 0.

In Example 7.5 we encountered case 1 and observed that2(Tn(a

,)) need not grow

at most polynomially in this case.

Theorem 7.6. In cases 2 and 3, the condition numbers 2(Tn(a

,)) grow at

most polynomially.

Proof. Let us consider case 3, the other case can be reduced to this case by passage to adjoints. We have

b(t) =a(t),= (t,)(t,)c(t); t2T;

where2T, 2T,6=, andchas neither poles nor zeros onT. Clearly,

wind(c(t);0) = lim

r!1,0

wind(b(rt);0) = lim

r!1,0

wind(a(rt);):

From Lemma 7.3 we deduce that wind(a(rt);) = 0 ifr 2(0;1) is suciently close

to 1. Hence, wind(c;0) = 0. From [7, Lemma 3 and Theorem 1] we therefore obtain that

kT ,1 n (b)

k

1=O(n

(20)

ELA

where" >0 can be chosen abitrarily. Since

kT ,1 n (b)

k 2

p

nkT ,1 n (b)

k 1;

it results that2(Tn(a

,)) =O(n 3=2+").

Notice that in Example 7.4 we have case 3 at=,1.

Appendix. In connection with Theorem 1.6, it is rather instructive to consider

2(Tn(a

,)) for a(z) =z+z

,1 and some selected values

2(a) = [,2;2]. We

rst note that

kT n(a

,)k 2

!ka,k

1= max 02

j2cos,j:

(36) SinceTn(a

,) is Hermitian, we have kT

,1 n (a

,)k 2=%(T

,1 n (a

,)) = 1=(n);

(37)

where(n) is given by (28). From (36) and (37) we get

2(Tn(a

,))

ka,k 1

(n) : (38)

Suppose rst that= 2. Then

(n) = min

1jn 2cos

j n+ 1,2

= 2

,2cos

n+ 1

2

n2;

and sinceka,2k

1= 4, we deduce from (38) that

2(Tn(a

,2))4 2=n2:

Analogously,

2(Tn(a+ 2)) 4

2=n2:

We remark that the last two relations are already in Grenander and Szego's book [13]. Now let= 2cosy withy2(0;). We have

2cos

j

n+ 1,2cosy = 4

sin

2

j n+ 1,

y

sin

2

j n+ 1 +y

:

(39)

Supposey==l=mis rational. Clearly, we can assume thatm2. In the case

wheren+ 10(modm), we have(n) = 0. Hence,

2(Tn(a

,)) =1 if n+ 10(modm):

If n+ 1 60(modm), there is a unique k 2 f1;2;:::;[m=2]gsuch that l(n+ 1) k(modm). A little thought reveals that

(n)4

2m(nk+ 1) sin2

l m+ml

2k m

sinl_m

1

(21)

ELA

whence

2(Tn(a

,))

ka,k 1

2k sin(l=mm )n: Consequently, the sequencef

2(Tn(a ,))g

1

n=1 can be divided into the subsequence f1;1;:::gand a nite number of subsequencesf

2(T n

(k ) i

(a,))gfor which

2(T n

(k ) i

(a,))C kn

(k ) i

with specic constantsCk. In particular,

2(Tn(a

,)) =1 for n,1(modm);

2(Tn(a

,))'n for n6,1(modm):

Finally, supposey=is irrational. Then, by (39),

2(Tn(a

,))<1 for all n1:

Ify=is algebraic of degree(2), then Liouville's theorem ensures the existence of

a constantc >0 such that

y ,

j n+ 1

c

(n+ 1)

for alln1. Thus, in this \tame" case we still have the estimate

2(Tn(a

,))Cn

for all n 1

(40)

with some constantC <1independent ofn. The derivation of (40) from Liouville's

theorem can also be found in Berg's book [1, pp. 105-106].

By a theorem of Dirichlet (see, e.g., [15, p. 4]), for every (not necessarily algebraic) irrational numbery=there exists a sequencen1< n2< n3< :::of natural numbers

such that the inequality 0<

y ,

jk

nk+ 1 <

1 (nk+ 1)

2

has a solutionjk

2f1;2;:::;n k

g. This implies that

2(Tn k(a

,))Dn 2 k

(41)

with some constantD2(0;1) which does not depend onn k.

Fazit: ify=is irrational, there is always a subsequencefn k

gsuch that (41) holds;

(22)

ELA

REFERENCES

[1] L. Berg,Lineare Gleichungssysteme mit Bandstruktur, Deutscher Verlag der Wissenschaften, Berlin 1986.

[2] R. Bhatia,Matrix Analysis, Springer-Verlag, New York 1996.

[3] A. Bottcher,Status report on rationally generated block Toeplitz and Wiener-Hopf determi-nants, Unpublished manuscript, 34 pages, 1989 (available from the author on request). [4] A. Bottcher, On the approximation numbers of large Toeplitz matrices, Doc. Math., 2:1{29,

1997.

[5] A. Bottcher and S. Grudsky, On the condition numbers of large semi-denite Toeplitz matrices,

Linear Algebra Appl., 279:285{301, 1998.

[6] A. Bottcher, S. Grudsky, and B. Silbermann, Norms of inverses, spectra, and pseudospectra of large truncated Wiener-Hopf operators and Toeplitz matrices,New York J. Math., 3:1{31, 1997.

[7] A. Bottcher and B. Silbermann, The asymptotic behavior of Toeplitz determinants for gener-ating functions with zeros of integral orders,Math. Nachr., 102: 79{105, 1981.

[8] A. Bottcher and B. Silbermann,Analysis of Toeplitz Operators, Akademie-Verlag, Berlin 1989 and Springer Verlag, Berlin, Heidelberg, New York 1990.

[9] A. Bottcher and B. Silbermann,Introduction to Large Truncated Toeplitz Matrices, Springer-Verlag, New York 1998.

[10] K.M. Day, Toeplitz matrices generated by the Laurent series expansion of an arbitrary rational function,Trans. Amer. Math. Soc., 206:224{245, 1975.

[11] K.M. Day, Measures associated with Toeplitz matrices generated by the Laurent expansion of rational functions,Trans. Amer. Math. Soc., 209:175{183, 1975.

[12] I. Gohberg and I.A. Feldman,Convolution Equations and Projection Methods for Their Solu-tion, Amer. Math. Soc., Providence, RI, 1974 [Russian original: Nauka, Moscow 1971]. [13] U. Grenander and G. Szego, Toeplitz Forms and Their Applications, University of California

Press, Berkeley and Los Angeles 1958.

[14] I.I. Hirschman, Jr., The spectra of certain Toeplitz matrices, Illinois J. Math., 11:145{159, 1967.

[15] E. Hlawka, J. Schoissengaier, and R. Taschner, Geometric and Analytic Number Theory, Springer-Verlag, Berlin, Heidelberg, New York 1991.

[16] M. Kac, W.L. Murdock, and G. Szego, On the eigenvalues of certain Hermitian forms, J. Rational Mech. Anal., 2:767{800, 1953.

[17] S.V. Parter, Extreme eigenvalues of Toeplitz forms and applications to elliptic dierence equa-tions,Trans. Amer. Math. Soc., 99:153{192, 1966.

[18] S. Roch and B. Silbermann, Index calculus for approximation methods and singular value decomposition,J. Math. Anal. Appl., 225:401{426, 1998.

[19] M. Rosenblatt, Some purely deterministic processes,J. Math. Mech., 6:801-810, 1957. [20] P. Schmidt and P. Spitzer, The Toeplitz matrices of an arbitrary Laurent polynomial,Math.

Scand., 8:15{38, 1960.

[21] S. Serra, On the extreme eigenvalues of Hermitian (block) Toeplitz matrices, Linear Algebra Appl., 270:109{129, 1997.

[22] S. Serra, How bad can positive denite Toeplitz matrices be?, Talk given at the conference \Fourier Analysis and Applications", Kuwait University, May 1998.