Invertibility of a class of Toeplitz operators over the half plane [Elektronische Ressource] / vorgelegt von Vladimir Vasilyev

Sujets

• Mathematics

Informations

• Publié par: technische_universitat_chemnitz
• Publié le: 01 janvier 2006
• Nombre de lectures: 18
• Langue: English

Extrait

Invertibility of a Class of Toeplitz
Operators over the Half Plane
von der Fakultat fur Mathematik
der Technischen Universitat Chemnitz
genehmigte
Dissertation
zur Erlangung des akademischen Grades
Doctor rerum naturalium
(Dr. rer. nat.)
vorgelegt von M.Sc. Vladimir A. Vasilyev
geboren am 24. August 1978 in Stavropol (Russland)
eingereicht am 28.09.2006
Gutachter: Prof. Dr. B. Silbermann, TU Chemnitz
Prof. Dr. V.S. Rabinovich,
Instituto Politecnico Nacional, Mexico
Prof. Dr. F.-O. Speck, Instituto Superior Tecnico, Portugal
Tag der Verteidigung: 07.02.2007Bibliographic description
Vasilyev, Vladimir Alexandrovich
Invertibility of a Class of Toeplitz Operators over the Half Plane
Dissertation (in English), 100 pages, Technical University of Chemnitz,
Faculty of Mathematics, Chemnitz, 2006.
Abstract
This paper is concerned with invertibility and one-sided invertibility of Toeplitz operators
over the half plane, whose symbols admit homogenous discontinuities, and with stability
of their pseudo nite sections.
The invertibility of this class of Toeplitz operators is studied using the related algebra
involving certain composition operators. The related stability problem plays here an im-
portant role. The invertibility criterium is given in terms of invertibility of a family of one
dimensional Toeplitz operators. The stability criterium for nite sections for the related
algebra is proved, and then used to get the stability of pseudo nite sections of Toeplitz
operators over the half plane.
The key observation to get one-sided invertibility criterium is building of a special function
which models the discontinuities of the original generating function. The form of this
function reveals the deep connections to the above mentioned related algebra. The one-
sided invertibility criterium is given it terms of constraints on the partial indices of certain
Toeplitz operator valued function.
Key words
Toeplitz operator, convolution operator, Toeplitz operator over the half plane, homoge-
nous discontinuities, approximate identities, stability, Banach algebra.Contents
1 Introduction 5
2 Preliminary Results 9
2.1 Banach algebras and their ideals . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Spaces of functions and sequences . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Finite sections of Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Toeplitz operators over the half plane and their pseudo nite sections . . . 15
2.6 Terminating indices of a Toeplitz operator . . . . . . . . . . . . . . . . . . 15
2.7 Sequences of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8 A local principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 A Related Stability Problem 18
3.1 Composition operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 The related algebra of sequences of functions . . . . . . . . . . . . . . . . . 19
3.3 The of operator sequences . . . . . . . . . . . . . . . . . . . 20
3.4 Stability in the related algebra . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Finite sections in the related algebra . . . . . . . . . . . . . . . . . . . . . 25
3.6 Description of local algebras of nite sections . . . . . . . . . . . . . . . . . 30
3.6.1 Scalar case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.6.2 Matrix case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.7 Stability of nite sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Functions admitting homogeneous discontinuities 38
4.1 Basic facts for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38N
4.2 Properties of function a(;t) when a2 and t2 T is xed . . . . . . . . 43N
4.3 Properties of functionsa^ in comparison witha^ when (s;t) belongs(s;t) (s ;t )0 0
to some small pinned neighborhood of (s ;t ) . . . . . . . . . . . . . . . . 440 0
04.4 A special representation for functions from . . . . . . . . . . . . . . . . 46N
14.4.1 Some properties of the operator C . . . . . . . . . . . . . . . . . 46r
4.4.2 A special representation with the help of composition operators . . 50
5 Invertibility of a Toeplitz operator over the half plane 54
5.1 Invertibility of operator valued function A(t) =T(a ) . . . . . . . . . . . . 57t
5.2 Certain C -algebra of operator valued functions . . . . . . . . . . . . . . . 58
5.3 Operator valued function A(t) =T(a ) belongs toB . . . . . . . . . . . . . 59t
5.4 Invertibility in algebraB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.5 Local representative of (T(a ) +J ) +I . . . . . . . . . . . . . . . . . . . 63t t j
tj
5.6 The structure of (B =J )=I . . . . . . . . . . . . . . . . . . . . . . . . . 65t jN
5.7 Invertibility of (T(a ) +J ) +I . . . . . . . . . . . . . . . . . . . . . . . 67t t 0
5.8 Criterium of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
36 One-sided invertibility 69
6.1 Squeezing function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Auxiliary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
06.3 Special function from . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.4 One-sided invertibility of operator valued function generated by special
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.5 Additional lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.6 Criterium of one-sided invertibility . . . . . . . . . . . . . . . . . . . . . . 77
7 Pseudo nite sections of a Toeplitz operator over the half plane 83
7.1 An algebra of nite sections . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.2 Invertibility offT (a )g +J +I . . . . . . . . . . . . . . . . . . . . . . . 88n t t j
7.3 Stability of nite sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8 Conclusions and Outlook 91
References 92
Selbstandigkeitserklarung (in Deutsch/in German) 95
Theses 96
41 Introduction
The Wiener-Hopf integral operators and the closely connected Toeplitz operators have
been intensively studied since 1950. These operator classes are of great interest from the-
oretical as well as from an applicative point of view. But we have to note that the multi-
dimensional case has been incomparably less studied than the one dimensional one. The
reasons for that are that in the multidimensional case topological questions are brought
to the forefront and problems appear, which do not play a signi can t role in the one
dimensional case.
In this dissertation we will mostly deal with Toeplitz operators over the half plane.
2Let T denote the unit circle in the complex plane, by T := TT we denote the torus. Let
1 2L (T ) denote the Banach space of equivalence classes of measurable N N matrixNN
2valued functions which are integrable on T . Let Z be the set of integer numbers. De ne
1 2for k = (k ;k )2 ZZ the Fourier coe cien ts of function a2L (T ) by1 2 NNZ Z
1 ds dt
a = a(s;t) :k 2 k +1 k +11 24 is it
T T
1 2Denote by L (T ) the Banach space of equivalence classes of measurable essentiallyNN
2bounded NN matrix valued functions on T .
2Let Z be the set of non-negative integer numbers. By l (Z Z) we denote the set+ +N
of square summable sequences of complex vectors withN components indexed by the set
1 2 2 2(Z Z). Fora2L (T ) let the operatorT (a) :l (Z Z)!l (Z Z) be given+ + + +NN N N
by the in nite matrix
T (a) =fa g :+ j k j;k2(Z Z)+
1 2It is well-known that T (a) is linear and bounded if and only if a 2 L (T ). The+ NN
operator T (a) is called the Toeplitz operators over the half plane.+
Such multidimensional Toeplitz op with continuous symbols were studied in
1960 by L.S. Goldenstein and I.Z. Gohberg who were the rst to prove the su ciency of
the following theorem (in [18]): P
1 2Theorem 1.1. Let a 2 L (T ) be such that ja j < 1. The operator T (a) isn +n2ZZ
invertible from at least one side if and only if
2a(s;t) = 0 for all (s;t)2 T :
If this condition is satis e d the operator T (a) is invertible, invertible from the left, in-+
vertible from the right if the number
1 i’ 2 = [arga(e ;t )] (t 2 T)0 0 0’=02
is respectively equal to zero, greater or less than zero. Note that the de nition of does0
not depend on t 2 T.0
5
6Later in 1967 L.S. Goldenstein proved the necessity (in [17]).
In articles [17] and [19] L.S. Goldenstein has also studied the stability of pseudo- nite
sections
T (a) =fa g ;n; j k j;k2([0;n]Z)
where [0;n] =f0; 1; 2;:::;ng, and proved the following theorem:
Theorem 1.2. Let a be as in Theorem 1.1. The sequence fT (a)g is stable, i.e. theren; n
exist m such that the operators T (a) are invertible for n>m andn;
1 sup T (a) <1;n;
n>m
if and only if the operator T (a) is invertible.+
As far as we know there are no analogous results for the matrix case even when theP
generating function a belongs to the Wiener algebra (i.e. ka k<1).nn2ZZ
In the discontinuous case generating functions which can be represented as a ten-
sor product of piecewise continuous functions were considered in [2]. Using the bilocal
Fredholm theory as in the quarter plane case in [1] the following the