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

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Factorization theory for Toeplitz plus Hankel operatorsand singular integral operators with ipVon der Fakult at fur Mathematik der Technischen Universtit at ChemnitzgenehmigteH a b i l i t a t i o n s s c h r i f tzur Erlangung des akademischen GradesDoctor rerum naturalium habilitatus(Dr. rer. nat. habil.)vorgelegtvon Dr. rer. nat. Torsten Ehrhardtgeboren am 25.9.1971 in Karl-Marx-Stadteingereicht am 12.8.2003Gutachter: 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: 5.7.2004http://archiv.tu-chemnitz.de/pub/2004/0124Torsten EhrhardtFactorization theory for Toeplitz plus Hankel operators and singular integral opera-tors with ipHabilitationschrift,Fakult atfur Mathematik,TechnischeUniversit atChemnitz,2004,ii+153 Seiten, http://www.archiv.tu-chemnitz.de/pub/2004/01242000 MSC: 47B35, 47A68Schlagw orter: Wiener-Hopf-Faktorisierung,Toeplitz-Operator,Hankel-Operator,sin-gul arer Integraloperator, FredholmtheorieKey words: Wiener-Hopf factorization, Toeplitz operator, Hankel operator, singularintegral operator, Fredholm theoryPrefaceInthisthesisweestablishafactorizationtheoryforToeplitzplusHankeloperatorsandfor singular integral operators with ip.

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Published 01 January 2004
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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 what follows, let us rst recall the basic facts of the factorization theory for
p NToeplitz operators T(c) acting on (H (T)) . It is well known that singular integral
operators (1.4) can be reduced to Toeplitz operators, which implies that the same
factorizationtheoryappliestosingularintegraloperatorsaswell. Standardreferences
for factorization theory are the monographs [11, 20].
First of all, one can classify factorizations into di erent types. The probably
simplest and historically rst considered one is given by the notion of Wiener-Hopf
∞factorizationinaBanachalgebra. LetB beaBanachalgebracontainedinL (T)and
containing the trigonometric polynomials, and suppose that the Riesz projection P
is bounded onB. Typical examples of such Banach algebras are the Banach algebras
∞of H older continuous functions on T or the Wiener algebra. Let B = B∩H (T)+
NN∞and B =B∩H (T). A matrix function c∈B is said to admit a Wiener-Hopf
factorization in B if it can be written as
c(t) = c (t) ( t)c (t), t∈T,+
NN NNwhere c ∈GB , c ∈GB , and ( t) is of the form+ +
κ κ1 N ( t) = diag(t ,...,t ) (1.5)
with κ ,...,κ ∈ Z being the so-called partial indices. The connection between1 N
Wiener-Hopf factorization and Fredholm properties of Toeplitz operators is given by
NNthe fact that if c ∈ B admits a Wiener-Hopf factorization in B, then T(c) is a
p NFredholm operator on (H (T)) and the defect numbers are given by
X X
dimkerT(c) = κ , dimkerT(c) = κ . (1.6)k k
κ <0 κ >0k k
Notice that this statement holds for arbitrary parameters 1<p<∞.
∞ NNForgeneralfunctionsc∈ (L (T)) , thenotionofWiener-Hopffactorizationin
a Banach algebra is not suitable, and another notion has therefore been introduced,
which allows the factors c and c and their inverses to be unbounded functions. Let+
∞ NN1 < p <∞ and 1/p+1/q = 1. A matrix function c∈ (L (T)) is said to admit
pa Fredholm factorization in L (T) if it can be written in the form
c(t) = c (t) ( t)c (t), t∈T,+
with ( t) given by (1.5) such that the following conditions are satis ed:
NN 1 NNp q(i) c ∈ (H (T)) , c ∈ (H (T)) .
q NN 1 p NN(ii) c ∈ (H (T)) , c ∈ (H (T)) .+ +
61 11(iii) The linear operator M(c )PM(c ), which is de ned on the linear space+
Nof allC -valued trigonometric polynomials, can be continued by continuity to
p Na linear bounded operator acting on (L (T)) .
Theimportanceofthisde nitionforToeplitzoperatorsisrevealedbythefundamental
∞ NNfact that for c ∈ (L (T)) the Toeplitz operator T(c) is Fredholm on the space
p N p(H (T)) if and only if the function c admits a Fredholm factorization in L (T).
In this case the defect numbers are given by the formulas (1.5). In particular, the
p NToeplitz operatorT(c) is invertible on (H (T)) if and only if it admits a generalized
pfactorization in L (T) with all partial indices being equal to zero. This result has
been proved rst by Simonenko [26].
ThenotionofFredholmfactorizationhasbeenstudiedinfurtherdetailbyClancey
and Gohberg [11], where it has been called generalized factorization. By Litvinchuk
and Spitkovsky [20] the same notion has been referred to as -factorization.
A factorization where merely conditions (i) and (ii), but not necessarily condition
p(iii)isful lledwillbereferredtoasa weak factorization inL (T). Weakfactorizations
has been studied in detail in the monograph [20]. The advantage of considering weak
factorizations is that the conditions are easy to verify, while on the other hand they
already ensure a certain kind of uniqueness of the factorization.
Condition (iii) appearing in the de nition of a Fredholm factorization seems in
general not easy to verify. However, there exists an equivalent condition related
pto the boundedness of the singular integral operator S on weighted L -spaces. This
conditionis, inturn, equivalentinthescalarcase(N =
1)totheHunt-MuckenhouptWheeden condition (or A -condition) [16] and in the matrix case (N > 1) to a muchp
more complicated generalization of the A -condition
[28].p
IthastobeemphasizedthatbyusingthefactorizationapproachtostudytheFredholm properties of Toeplitz operators, one encounters severe di culties. Namely, in
general,itisunavoidabletofactorthegivenmatrixfunctionexplicitly. Unfortunately,
only for certain classes of functions a factorization can be constructed explicitly and
the defect numbers can be computed. There exists a vast and quite heterogeneous
literature devoted to explicit factorization techniques. Each new method for explicit
factorization represents a huge progress, which can usually be achieved only with
enormous e orts. Explicitly factorable matrix functions include, for instance,
rational matrix functions, upper triangular matrix functions, so-called Daniele-Khrapkov
matrix functions and piecewise constant matrix functions. We will not make an
attempt here to give an overview but just refer to [11, 20, 13, 14] for some examples,
further information and references.
On the other hand, the di culty of explicit factorization seems at least to some
extent to steem from the original Fredholm problem itself. One theoretical
explanation for the di culties can be seen in the fact that the defect numbers (as the partial
indices) are instable under small perturbations. But it should also be noted that in
many non-trivial cases in which one has been able to compute the defect numbers
this has been done by means of factorization.
7It is also necessary to mention that there exists an important class of functions,
namely,piecewise continuous functionsontheunitcircle,forwhichaFredholmtheory
canbeestablishedwithoutresortingtofactorizationtheory. Thisholdstrue,however,
only as far as necessary and su cient criteria for the Fredholmness of T(c) and the
computation of the Fredholm index are concerned. In regard to the defect numbers
themselves, this approach does not yield any information in the matrix case (N > 1).
Still in the scalar case (N = 1), the defect numbers can be computed by means of
a result due to Coburn. Hence invertibility for Toeplitz operators with piecewise
continuous generating functions can be settled without making use of factorization
theory in the scalar case, but not in the matrix case.
After these introductory remarks concerning the factorization theory for Toeplitz
operators, which, as we already said, apply also to singular integral operators (1.4),
let us proceed with recalling the known results for Toeplitz plus Hankel operators
and singular integral operators with ip (1.2) and (1.3) and then continuing with
explaining the content of the dissertation.
For Toeplitz plus Hankel operators and singular integral operators with ip
generated by piecewise continuous matrix functions, necessary and su cient criteria for
the Fredholmness and a formula for the Fredholm index are known or can easily be
obtained from known results [24, 25]. This holds true even for more general singular
integral operators with di erent types of ip operators and considered on more
general curves. Many of the material can be found in the monograph by Kravchenko and
Litvinchuk [18]. However, concerning the computation of the defect numbers and in
regard to necessary and su cient criteria for the invertibility the approach followed
there does not yield any information, not even in the scalar case (N = 1).
On the other hand, it has long been known that Toeplitz plus Hankel operators
can be related to singular integral operator (1.4) in some, though unsatisfactory,
way. In order to establish this relationship one has to consider the Toeplitz plus
Hankel operators T(a) + H(b) and T(a) H(b) simultaneously. For instance, the
invertibility of both these operators is equivalent to the invertibility of an associated
singular integral operator of the form (1.4), to which one can apply the well known
factorizationtheory. Thedrawbackofthisapproachisthatonlysu cientinvertibility
criteria for T(a)+H(b) can be derived.
The same kind of observation applies to singular integral operators with ip (1.2)
and (1.3), which can also be related to singular integral operators (1.4) after taking
into account a further, unwanted singular integral operator with ip. Again, only
su cient criteria for invertibility can be derived.
Despite of the hint which one might conceive from this observations, nobody
seemed to have tried to approach Toeplitz plus Hankel operators or singular integral
operators with ip directly with factorization theory, nor does there exist – to the
bestoftheauthor’sknowledge–aninvertibilitytheoryforsuchoperatorsoraclassof
them at all. As often in such situations, where nobody believes that further progress
can be made, a certain amount of luck is necessary in order to persuade oneself to
8