Factorization theory for Toeplitz plus Hankel operators and singular integral operators with flip [Elektronische Ressource] / von Torsten Ehrhardt

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• Mathematics

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• Publié par: technische_universitat_chemnitz
• Publié le: 01 janvier 2004
• Nombre de lectures: 11
• Langue: English

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Factorization theory for Toeplitz plus Hankel operators
and singular integral operators with ip
Von der Fakult at fur Mathematik der Technischen Universtit at Chemnitz
genehmigte
H a b i l i t a t i o n s s c h r i f t
zur Erlangung des akademischen Grades
Doctor rerum naturalium habilitatus
(Dr. rer. nat. habil.)
vorgelegt
von Dr. rer. nat. Torsten Ehrhardt
geboren am 25.9.1971 in Karl-Marx-Stadt
eingereicht am 12.8.2003
Gutachter: Prof. Dr. rer. nat. habil. Bernd Silbermann (Chemnitz)
Prof. Dr. rer. nat. habil. Frank-Olme Speck (Lissabon)
Prof. Dr. rer. nat. habil. Ilya Spitkovsky (Williamsburg)
Tag der Verleihung des akademischen Grades: 5.7.2004
http://archiv.tu-chemnitz.de/pub/2004/0124Torsten Ehrhardt
Factorization theory for Toeplitz plus Hankel operators and singular integral
operators with ip
Habilitationschrift,Fakult atfur Mathematik,TechnischeUniversit atChemnitz,2004,
ii+153 Seiten, http://www.archiv.tu-chemnitz.de/pub/2004/0124
2000 MSC: 47B35, 47A68
Schlagw orter:
Wiener-Hopf-Faktorisierung,Toeplitz-Operator,Hankel-Operator,singul arer Integraloperator, Fredholmtheorie
Key words: Wiener-Hopf factorization, Toeplitz operator, Hankel operator, singular
integral operator, Fredholm theoryPreface
InthisthesisweestablishafactorizationtheoryforToeplitzplusHankeloperatorsand
for singular integral operators with ip. These operators are considered with matrix
symbolsandarethoughtofactingonthevector-valuedanaloguesoftheHardyspaces
p pH (T) and Lebesgue spaces L (T) with 1<p<∞.
A factorization theory for pure Toeplitz operators and singular integral operators
without a ip is known since decades and provides necessary and su cient conditions
for the Fredholmness of such operators along with formulas for their defect
numbers. In particular, the invertibility of such operators is essentially equivalent to the
existence of a certain type of Wiener-Hopf factorization.
It has been an open question whether some kind of factorization theory for the
moregeneralclassesofToeplitzplusHankeloperatorsandsingularintegraloperators
with ip exists at all. In this thesis it is shown that the answer is a rmative and the
corresponding theory is developed.
Itturnsoutthatthefactorizationwhichisappropriateforthesesgeneralclassesof
operatorsisofacompletelydi erentkind.
SeveralnotionsasthoseofweakandFredholm asymmetric factorization as well as antisymmetric factorization are introduced
and studied. A Fredholm theory including the computation of the defect numbers
is established from which an invertibility theory can be derived. Connections with
the Hunt-Muckenhoupt-Wheeden (or A -condition) are made. Several illustratingp
examples and applications are given as well.
Acknowledgments. IwishtoexpressmyspecialgratitudetoBerndSilbermann
who has accompanied me along my scienti c career for almost fteen year. Working
in his research group in Chemnitz has not only been intellectually very fruitful and
stimulating, but it has also been a great pleasure. I am also deeply indebted to Ilya
Spitkovsky and Frank-Olme Speck for their e orts in reading this habilitation thesis
and for their many useful remarks.
Chemnitz, August 2004 Torsten Ehrhardt
12Contents
1 Introduction 5
2 Basic de nitions and results 15
2.1 Basic de nitions and notation . . . . . . . . . . . . . . . . . . . . . . 15
2.2 General operator theoretic preliminaries . . . . . . . . . . . . . . . . 18
2.3 Toeplitz operators and Hankel operators . . . . . . . . . . . . . . . . 23
2.4 Toeplitz plus Hankel operators . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Singular integral operators . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Wiener-Hopf factorization in Banach algebras . . . . . . . . . . . . . 29
2.7 Factorization theory and the Fredholmness of Toeplitz operators . . . 33
p2.8 Matrix weighted L -spaces and singular integral operators . . . . . . 37
3 Toeplitz plus Hankel operators and singular integral operators with
ip 43
3.1 The classical approach to Toeplitz plus Hankel operators . . . . . . . 44
3.2 The reduction of general Toeplitz plus Hankel operators . . . . . . . . 46
3.3 Then of singular integral operators with ip . . . . . . . . . 48
3.4 The operatorsM (a) andN (a) . . . . . . . . . . . . . . . . . . . . 50w w
3.5 Duality betweenM (a) andN (b) . . . . . . . . . . . . . . . . . . . 57w w
3.6 The operatorsM(a) andN(a) . . . . . . . . . . . . . . . . . . . . . 62
3.7 The operatorsM(a) andN(b) with PC-symbols . . . . . . . . . . . 65
4 Factorizations in a Banach algebra 69
4.1 Antisymmetric factorization in a Banach algebra . . . . . . . . . . . . 70
4.2 Asymmetric factorization in a Banach algebra . . . . . . . . . . . . . 76
4.3 fation in a Banach algebra and Fredholmness . . . 79
4.4 factorization in Banach algebras of continuous functions 88
4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.6 Applications to general Toeplitz plus Hankel operators . . . . . . . . 96
5 On the kernel and cokernel of Toeplitz plus Hankel operators 101
n n5.1 Basic properties ofM (at ) andN (bt ) . . . . . . . . . . . . . . 102w w
n n5.2 The kernels ofM (at ) andN (bt ). . . . . . . . . . . . . . . . . 105w w
n n5.3 Duality ofM (at ) andN (bt ) . . . . . . . . . . . . . . . . . . . 112w w
3n n5.4 The cokernels ofM (at ) andN (bt ) . . . . . . . . . . . . . . . 115w w
5.5 Auxiliary results for factorization . . . . . . . . . . . . . . . . . . . . 120
5.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6 Toeplitz plus Hankel operators and weak factorization 135
6.1 The notion of weak asymmetric factorization . . . . . . . . . . . . . . 136
6.2 Necessary and su cient conditions for the existence of a weak
asymmetric factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.3 Weak antisymmetric factorizations . . . . . . . . . . . . . . . . . . . 154
6.4 On the uniqueness of the weak factorizations . . . . . . . . . . . . . . 161
6.5
Relationofweakantisymmetricfactorizationstoantisymmetricfactorizations in a Banach algebra . . . . . . . . . . . . . . . . . . . . . . . 166
6.6 Examples of weak factorizations . . . . . . . . . . . . . . . . . . . . . 169
6.7 Applications to general Toeplitz plus Hankel operators . . . . . . . . 171
7 Toeplitz plus Hankel operators and Fredholm factorization 181
7.1 Fredholmness ofM (a) andN (b) . . . . . . . . . . . . . . . . . . . 181w w
7.2 Proof of Theorem 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.3 Asymmetric Fredholm factorization . . . . . . . . . . . . . . . . . . . 197
7.4 The adjoints of (a) and (b) . . . . . . . . . . . . . . . . . . . . 200w w
7.5 On the continuability of (a) and (b) . . . . . . . . . . . . . . . 205w w
7.6 Proof of Proposition 7.28 and Theorem 7.30 . . . . . . . . . . . . . . 209
7.7 Duality and the continuability of F and F . . . . . . . . . . . . . . . 217
7.8 The continuability ofF andF and its relation tothe singular integral
operator on [ 1,1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
7.9 Pseudoinverses ofM (a) andN (b) . . . . . . . . . . . . . . . . . . 226w w
7.10 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
A Fredholm theory in case of piecewise continuous functions 233
p NNA.1 Fredholm theory for operators from the algebraS (PC ) . . . . . 234
NNA.2 Fr for Toeplitz plus Hankel operators with PC -
symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
A.3 Fredholm theory for particular Toeplitz plus Hankel operators with
NNPC -symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
Bibliography 243
Notation index 247
Symbol index 249
Theses 251
Erklarung 253
4Chapter 1
Introduction
Thegoalofthisdissertationistodevelopafactorizationtheorywhichallowstostudy
Toeplitz plus Hankel operators
T(a)+H(b) (1.1)
and singular integral operators with ip
PM(a)+PM(b)J +QM(c)J +QM(d) (1.2)
and
M(a)P +M(b)JP +M(c)JQ+M(d)Q. (1.3)
Here T(a) and H(b) stand for Toeplitz and Hankel operator acting on the vector
p Nvalued Hardy space (H (T)) on the unit circle T, 1 < p < ∞, with generating
∞ NNfunctions belonging to (L (T)) . The above singular integral operators with ip
p Nare thought of acting on the spaces (L (T)) and contain multiplication operators
∞ NNwith generating functions a,b,c,d∈ (L (T)) . The operators P = (I +S)/2 and
Q = (I S)/2 are the Riesz projections, S is the singular integral operator onT,
1 1and J stands for the ip operator ( Jf)(t) = t f(t ), t∈T. Notice that J is a ip
operator which changes the orientation of the underlying curveT.
Factorization theory is a well establish method for studying Fredholm properties
of Toeplitz operators
T(a),
singular integral operators
PM(a)+QM(b) and M(a)P +M(b)Q, (1.4)
andevenmuchmoregeneralToeplitzoperatorsandsingularintegraloperatorsrelated
to di erent curves and weighted spaces. By studying Fredholm properties of an
operator A we mean not only to establish necessary and su cient criteria for the
Fredholmness of A, but we also want to obtain formulas for the defect numbers
dimkerA and dimkerA .
5Only the knowledge of these numbers allows in general to establish necessary and
su cient criteria for the invertibility of A.
In