Convolution Operators and Factorization of Almost Periodic Matrix Functions

461 Pages·2002·17.67 MB·English

by  Albrecht Böttcher

Operator Theory: Advances and
Applications
Vol.131
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Convolution Operators and
Factorization of Almost
Periodic Matrix Functions
Albrecht Böttcher
Yuri I. Karlovich
llya M. Spitkovsky
Springer Basel AG
Authors:
Albrecht Böttcher Yuri I. Karlovich
Faculty of Mathematics Department of Mathematics
Technical University Chemnitz CINVESTAV of the I.P.N.
09107 Chemnitz P.O. Box 14-740
Germany 07000 Mexico D.F.
e-mail: [e-mail] Mexico
e-mail: karlovic @math.cinvestav.mx
llya M. Spitkovsky
Department of Mathematics
College of William and Mary
P.O. Box 8795
Williamsburg, VA 23187-8795
USA
e-mail: [e-mail]
2000 Mathematics Subject Classification 47A68, 47B35
A CIP catalogue record for this book is available from the
Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Böttcher, Albrecht:
Convolution operators and factorization of almost periodic matrix functions
/ Albrecht Böttcher ; Yuri I. Karlovich ; llya M. Spitkovsky. - Basel ;
Boston ; Berlin : Birkhäuser, 2002
(Operator theory ; Vol. 131)
ISBN 978-3-0348-9457-9 ISBN 978-3-0348-8152-4 (eBook)
DOI 10.1007/978-3-0348-8152-4
ISBN 978-3-0348-9457-9
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,
specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms
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© 2002 Springer Basel AG
Originally published by Birkhäuser Verlag in 2002
Softcover reprint of the hardcover 1st edition 2002
Member of the BertelsmannSpringer Publishing Group
Printed on acid-free paper produced from chlorine-free pulp. TCF «>
Cover design: Heinz Hiltbrunner, Basel
ISBN 978-3-0348-9457-9
Contents
Preface ...................................................................... ix
1 Convolution Operators and Their Symbols
1.1 Banach and C* -Algebras .............................................. 1
1.2 Convolution Operators on the Line .................................... 4
1.3 Almost Periodic Symbols .............................................. 6
1.4 Kronecker's Theorem .................................................. 8
1.5 Semi-Almost Periodic Symbols ....................................... 14
1.6 Convolution Operators on a Half-Line ................................ 20
1.7 Convolution Operators on Finite Intervals .......................... " 21
2 Introduction to Scalar Wiener-Hopf Operators
2.1 Fredholm and Semi-Fredholm Operators ............................. 27
2.2 Two Basic Theorems ................................................ 29
2.3 Hankel Operators ................................................... 30
2.4 Continuous Symbols ................................................. 32
2.5 C + HOC Symbols ................................................... 35
2.6 PC Symbols ........................................................ 37
2.7 Mean Value and Bohr-Fourier Spectrum ............................. 41
2.8 Bohr's Theorem and Mean Motion.................................. 42
2.9 AP Symbols ........................................................ 45
3 Scalar Wiener-Hopf Operators with SAP Symbols
3.1 The Geometric Mean ................................................ 51
3.2 Sarason's Lemma .................................................... 55
3.3 Canonical Representatives ........................................... 56
3.4 Sarason's Theorem .................................................. 60
3.5 Index Formula ...................................................... 63
3.6 SAP + PCa Symbols ................................................ 66
4 Some Phenomena Caused by SAP Symbols
4.1 Local Nature of the Essential Spectrum ............................. 70
4.2 Prescribed Stars in the Essential Spectrum .......................... 73
4.3 Essential Spectra of Prescribed Global Shapes ....................... 77
4.4 Separated Almost Periodic Representatives .......................... 82
4.5 The Spectrum is Discontinuous ...................................... 85
4.6 Amplitude Modulation Preserves Fredholmness ...................... 88
4.7 Frequency Modulation Can Destroy Fredholmness ................... 89
VI Contents
5 Introduction to Matrix Wiener-Hopf Operators
5.1 General Remarks and Normal Solvability ............................ 93
5.2 Matrix-Valued C + HOO Symbols .................................... 96
5.3 Matrix-Valued PC Symbols ......................................... 98
5.4 The Gohberg-Krein Theorem ....................................... 102
5.5 Outlook............................................................ 104
6 Factorization of Matrix Functions
6.1 Hardy Spaces ...................................................... 107
6.2 Wiener-Hopf Factorization ......................................... 110
6.3 Almost Periodic Factorization ....................... , . . .. . .. . .. .. .. 114
6.4 Unitary Middle Factors ............................................ 115
7 Bohr Compactification
7.1 Some Commutative Harmonic Analysis ............................. 121
7.2 The Bochner-Fejer Operators. .. . .. .. . . .. . . .. . .. . .. . . . .. ... . . . .. . ... 124
7.3 Another Look at AP Factorization. .. . . .. .. .. .. . . . . . .. .. .. . . . . . . . .. 127
7.4 The Besicovitch Space ............................... , . . . . .. . . .. . ... 127
8 Existence and Uniqueness of AP Factorization
8.1 Uniqueness of AP Factorization.. .. . . . . .. . .. .. .. .. .. . .. .. .. .. .. .. .. 131
8.2 AP Indices and Geometric Mean................................... 134
8.3 Scalar Case ........................................................ 136
8.4 Periodic Matrix Functions .......................................... 142
8.5 An Invertible APW Polynomial Without APW Factorization ...... 145
8.6 Stabilty of AP Indices ............................................. 147
9 Matrix Wiener-Hopf Operators with APW Symbols
9.1 An Isomorphism Theorem for CO-Dynamical Systems.............. 155
9.2 Passage to the Besicovitch Space ................................... 159
9.3 Application of the Bochner-Phillips Theorem.. . .. .. . .. . . . . . . .. . .. .. 162
9.4 Invertibility of Operators with APW Symbols ....... , .. .. . .. .. .. ... 164
9.5 APW Symbols with Good Numerical Range. . . . . . . . . . . . . . . . . . . . . . .. 167
9.6 Hermitian Matrix Symbols in APW ................................ 169
9.7 One-Sided Invertibility of Operators with APW Symbols. . . . . . . . ... 174
10 Matrix Wiener-Hopf Operators with SAPW Symbols
10.1 Invertibility of the Almost Periodic Representatives ............... 181
10.2 Fredholmness of Operators with APW Symbols ................... 185
10.3 Reduction of SAP Symbols to PC Symbols ....................... 187
10.4 Fredholmness and Index of Operators with SAPW Symbols ....... 190
10.5 Two Results from Linear Algebra ................................. 193
10.6 Semi-Fredholm Theory ............................................ 196
Contents vii
11 Left Versus Right Wiener-Hopf Factorization
11.1 The Associated Operator ...................... . . . . . . . . . . . . . . . . . . .. 207
11.2 The Index of the Associated Operator ....... , . . . .. ... . . . .. . . . .. . .. 208
11.3 Introduction to Toeplitz Operators. " .. .. . . . . .. . .. . . .. .. . . . .. .. . .. 211
11.4 The Inequality for the Total Indices ............................... 212
11.5 Bounded Symbols with Prescribed Total Indices. . . .. .. .. . . . .. . . . .. 217
11.6 SAP Symbols with Prescribed Total Indices. . . . .. .. .. . .. . . . .. ... .. 220
12 Corona Theorems
12.1 The Corona Problem for AP and APW 227
12.2 The Arens-Singer Theorem. . . . . . . . . . . . .. .. .. . . . . . .. . . . . . . . . .. ... .. 230
12.3 Empty and Nonempty Coronas .................................... 236
13 The Portuguese Transformation
13.1 Our Tower of Babel ............................................... 243
13.2 Preliminary Observations ......................................... 246
13.3 The Idea of the Portuguese Transformation ....................... 248
13.4 A Special Corona Problem: Scalar Case ........................... 250
13.5 A Special Corona Problem: Matrix Case .......................... 254
13.6 One More Approach .............................................. 256
14 Some Concrete Factorizations
14.1 The One-Sided Case.............................................. 259
14.2 The Big Gap Case ................................................ 262
14.3 Scalar Binomials .................................................. 267
14.4 Commensurable Distances ......................................... 271
14.5 The Division Algorithm.. . . . . . . . .. .. .. .. . . . . . . . .. . . . .. . .. .. . .. .... 274
15 Scalar Trinomials
15.1 v + /j = A, /-L = 0 ................................................... 281
15.2 v + /j > A, /-L = 0 ................................................... 289
15.3 v + /j = A, /-L =J 0 ................................................... 294
15.4 v + /j > A, /-L =J 0 ................................................... 296
16 Toeplitz Operators
16.1 Muckenhoupt Weights ............................................ 301
16.2 Simonenko's Theorems ............................................ 304
16.3 Emergence of Horns ............................................... 306
16.4 Toeplitz Operators with PC Symbols ............................. 315
17 Zero-Order Pseudodifferential Operators
17.1 Fourier Multipliers ................................................ 323
17.2 Wiener-Hopf Operators with PC Symbols ......................... 326
17.3 The Symbol Calculus ............................................. 332
viii Contents
18 Toeplitz Operators with SAP Symbols on Hardy Spaces
18.1 Matrix-Valued APW Symbols .................................... . 337
18.2 Necessary Conditions for Semi-Fredholmness ..................... . 341
18.3 Saginashvili's Theorem ........................................... . 345
18.4 Invertibility of the Almost Periodic Representatives .............. . 347
18.5 Existence and Continuity of the Geometric Mean ................. . 349
18.6 The First Auxiliary Operator .................................... . 352
18.7 The Second Auxiliary Operator .................................. . 355
18.8 Matrix-Valued SAP Symbols .................................... . 357
18.9 Spaces with Power Weights ...................................... . 366
19 Wiener-Hopf Operators with SAP Symbols on Lebesgue Spaces
19.1 Multipliers from AP and SAP ................................... . 371
19.2 Invertibility of the Symbol ....................................... . 374
19.3 Invertibility of the Almost Periodic Representatives .............. . 376
19.4 Matrix-Valued APW Symbols .................................... . 378
19.5 The Theorem by Duduchava and Saginashvili .................... . 381
19.6 Matrix-Valued SAP Symbols .................................... . 384
20 Hankel Operators on Besicovitch Spaces
20.1 Bl, B2, Boo ..................................................... . 387
20.2 Norms of Matrix Hankel Operators on B2 ........................ . 389
20.3 Matrix Hankel Operators with AP Symbols ...................... . 391
20.4 Applications to Wiener-Hopf Operators .......................... . 393
21 Generalized AP Factorization
21.1 Definition of Generalized AP Factorization 395
21.2 Two Lemmas .................................................... . 397
21.3 Invertibility of Matrix Operators with AP Symbols ............... . 399
21.4 And Once More Matrix Operators with SAP Symbols ........... . 402
22 Canonical Wiener-Hopf Factorization via Corona Problems
22.1 Canonical Left W H Factorization and Corona Problems. . . . . . . . . .. 408
22.2 Separation Principle for Corona Problems ......................... 415
22.3 Explicit Formulas for Basic Corona Solutions ...................... 418
22.4 Canonical W H Factorization of Triangular Matrix Functions 423
23 Canonical APW Factorization via Corona Problems
23.1 General Results ................................................... 427
23.2 Binomial Periodic Corona Data ................................... 433
23.3 Trinomial Matrix Functions ....................................... 435
Bibliography ............................................................... 441
Index ...................................................................... 459
Preface
Many problems of the engineering sciences, physics, and mathematics lead to con
volution equations and their various modifications. Convolution equations on a
half-line can be studied by having recourse to the methods and results of the
theory of Toeplitz and Wiener-Hopf operators. Convolutions by integrable kernels
have continuous symbols and the Cauchy singular integral operator is the most
prominent example of a convolution operator with a piecewise continuous symbol.
The Fredholm theory of Toeplitz and Wiener-Hopf operators with continuous and
piecewise continuous (matrix) symbols is well presented in a series of classical and
recent monographs.
Symbols beyond piecewise continuous symbols have discontinuities of oscillating
type. Such symbols emerge very naturally. For example, difference operators are
nothing but convolution operators with almost periodic symbols: the operator
defined by (A<p)(x) = L: ak<P(x - Ak) is the convolution operator with the symbol
f(x) = L:akei>"kX. Moreover, a convolution operator on a finite interval is, in
a sense, equivalent to a convolution operator on the half-line whose symbol is a
2 x 2 oscillating matrix function: consideration of the convolution operator with
the symbol f(x) on the interval (0, A) leads to the convolution operator with the
matrix symbol
on the half-line (0,00). Notice that eVl(x) is oscillating even if f(x) is continu
ous. We finally mention that convolution operators with oscillating symbols have
properties that are not shared by operators with continuous or piecewise continu
ous symbols. Some basic phenomena of Toeplitz and Wiener-Hopf theory stay in
the dark when working with piecewise continuous symbols but come to light after
passage to, say, semi-almost periodic symbols. One illustration of this statement
is as follows. Let an be functions that converge uniformly to some function a. If
the functions an are continuous or piecewise continuous, then the spectra of the
Wiener-Hopf operators with the symbols an converge in the Hausdorff metric to
the spectrum of the Wiener-Hopf operator with the symbol a. Surprisingly, this
need not be true in case the functions an are semi-almost periodic.
x Preface
Convolution equations on the half-line can be solved once a Wiener-Hopf factoriza
tion of the symbol is available. In this book we investigate several factorizations of
almost periodic matrix functions. For instance, in case the function f(x) is almost
periodic, we are interested in representing G?)(x) in the form
where J.Ll,J.L2 are real numbers and G±(x) are invertible almost periodic matrix
functions such that the Bohr-Fourier spectra of G!l(X) and G::1(x) are contained
in [0,00) and (-00,0]' respectively. Such a factorization is called an AP factoriza
tion. The numbers J.Ll, J.L2 are referred to as the almost periodic indices of G~)..) (x).
Sole knowledge of the almost periodic indices tells us much about the properties of
the original convolution operator. If, in addition, the factors G _ (x) and G + (x) are
at our disposal, we have nearly complete command of the convolution equation.
The notion of AP factorization was introduced in the eighties. During the last
twenty years, AP factorization has become a relatively independent business.
Nowadays one knows large classes of symbols f(x) for which an AP factorization
of G~)..) (x) can be constructed explicitly. In the other direction, one has discovered
classes of very nice symbols f(x) such that G~)..)(x) has definitely no AP factor
ization. The border between these two extreme situations remains a rugged crest
with many open questions and mysteries.
The purpose of this book is to elucidate the close relationship between convolution
equations and AP factorization on the one hand and to acquaint the reader with
what is known about the AP factorization of several concrete classes of matrix
functions on the other. Our emphasis is both on the results and on the methods of
AP factorization. The majority of the results we will quote are accompanied with
full proofs. Moreover, many results are taken from the periodicals and are cited
with full proofs here for the first time. We suppose that knowledge of the basics
of functional, real, complex, and harmonic analysis will enable the interested and
motivated reader to go through the bulk of the book without serious difficulty.
A look at the table of contents provides an idea of what this book is actually about.
We have split the large amount of material into many small pieces, so that the book
has 23 chapters. We believe that this facilitates getting in the matter. Moreover,
each chapter has its own flavor: some of them are introductory surveys, others
are devoted to delightful stories, and part of the chapters are forced marches. The
pictures on the first pages of the chapters show the ranges of different semi-almost
periodic functions and are explained in Chapter 3.
It should not be hidden that, despite many genuine achievements made during the
last two decades, the edifice of AP factorization is still incomplete and that our
original optimism to finish this edifice in due time has deceased over the years.
We nevertheless think the time is ripe for this book. It is the first attempt of