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A few years aga the authors started a project of a book on the
theory of systems of one-dimensional singular integral equa tions
which was planned as a continuation of the monograph by one of the
authors and N. Ya. Krupnik ~~ concerning scalar equa tions. This
set of notes was initiated as a chapter dealing with problems of
factorization of matrix functions vis-a-vis appli cations to
systems of singular integral equations. Working systematically
onthischapter and adding along the way new points of view, new
proofs and results, we finally saw that the material connected with
factorizations is of independent interest and we decided to publish
this chapter as aseparate volume. In fact, because of recent
activity, the amount of material was quite large and we quickly
learned that we cannot cover all of the results in complete detail.
We have tried to include a represen tative variety of all kinds of
methods, techniques,results and applications. Apart of the current
work exposes results from the Russian literature which have never
appeared in English translation. We have also decided to reflect
some of the recent results which make interesting connections
between factorization of matrix functions and systems theory. The
field remains very active and many results and connec tions are
still not weIl understood. These notes should be viewed as a
stepping stone to further development. The authors hope that
sometime they will return to complete their original plan.
One of the basic interpolation problems from our point of view is
the problem of building a scalar rational function if its poles and
zeros with their multiplicities are given. If one assurnes that the
function does not have a pole or a zero at infinity, the formula
which solves this problem is (1) where Zl , " " Z/ are the given
zeros with given multiplicates nl, " " n / and Wb" " W are the
given p poles with given multiplicities ml, . . . ,m , and a is an
arbitrary nonzero number. p An obvious necessary and sufficient
condition for solvability of this simplest Interpolation pr- lern
is that Zj :f: wk(1~ j ~ 1, 1~ k~ p) and nl +. . . +n/ = ml +. . .
+m ' p The second problem of interpolation in which we are
interested is to build a rational matrix function via its zeros
which on the imaginary line has modulus 1. In the case the function
is scalar, the formula which solves this problem is a Blaschke
product, namely z z. )mi n u(z) = all = l~ (2) J ( Z+ Zj where [o]
= 1, and the zj's are the given zeros with given multiplicities mj.
Here the necessary and sufficient condition for existence of such
u(z) is that zp :f: - Zq for 1~ ]1, q~ n.
This paper is a largely expository account of the theory of p x p
matrix polyno mials associated with Hermitian block Toeplitz
matrices and some related problems of interpolation and extension.
Perhaps the main novelty is the use of reproducing kernel
Pontryagin spaces to develop parts of the theory in what hopefully
the reader will regard as a reasonably lucid way. The topics under
discussion are presented in a series of short sections, the
headings of which give a pretty good idea of the overall contents
of the paper. The theory is a rich one and the present paper in
spite of its length is far from complete. The author hopes to fill
in some of the gaps in future publications. The story begins with a
given sequence h_n" ... , hn of p x p matrices with h-i = hj for j
= 0, ... , n. We let k = O, ... ,n, (1.1) denote the Hermitian
block Toeplitz matrix based on ho, ... , hk and shall denote its 1
inverse H k by (k)] k [ r = .. k = O, ... ,n, (1.2) k II} . '-0 '
I- whenever Hk is invertible.
This two-volume work presents a systematic theoretical and
computational study of several types of generalizations of
separable matrices. The main attention is paid to fast algorithms
(many of linear complexity) for matrices in semiseparable,
quasiseparable, band and companion form. The work is focused on
algorithms of multiplication, inversion and description of
eigenstructure and includes a large number of illustrative examples
throughout the different chapters. The first volume consists of
four parts. The first part is of a mainly theoretical character
introducing and studying the quasiseparable and semiseparable
representations of matrices and minimal rank completion problems.
Three further completions are treated in the second part. The first
applications of the quasiseparable and semiseparable structure are
included in the third part where the interplay between the
quasiseparable structure and discrete time varying linear systems
with boundary conditions play an essential role. The fourth part
contains factorization and inversion fast algorithms for matrices
via quasiseparable and semiseparable structure. The work is based
mostly on results obtained by the authors and their coauthors. Due
to its many significant applications and the accessible style the
text will be useful to engineers, scientists, numerical analysts,
computer scientists and mathematicians alike.
This two-volume work presents a systematic theoretical and
computational study of several types of generalizations of
separable matrices. The main attention is paid to fast algorithms
(many of linear complexity) for matrices in semiseparable,
quasiseparable, band and companion form. The work is focused on
algorithms of multiplication, inversion and description of
eigenstructure and includes a large number of illustrative examples
throughout the different chapters.
The second volume, consisting of four parts, addresses the
eigenvalue problem for matrices with quasiseparable structure and
applications to the polynomial root finding problem. In the first
part the properties of the characteristic polynomials of principal
leading submatrices, the structure of eigenspaces and the basic
methods to compute eigenvalues are studied in detail for matrices
with quasiseparable representation of the first order. The second
part is devoted to the divide and conquer method, with the main
algorithms being derived also for matrices with quasiseparable
representation of order one. The QR iteration method for some
classes of matrices with quasiseparable of any order
representations is studied in the third part. This method is then
used in the last part in order to get a fast solver for the
polynomial root finding problem. The work is based mostly on
results obtained by the authors and their coauthors. Due to its
many significant applications and the accessible style the text
will be useful to engineers, scientists, numerical analysts,
computer scientists and mathematicians alike.
This book aims to present the theory of interpolation for rational
matrix functions as a recently matured independent mathematical
subject with its own problems, methods and applications. The
authors decided to start working on this book during the regional
CBMS conference in Lincoln, Nebraska organized by F. Gilfeather and
D. Larson. The principal lecturer, J. William Helton, presented ten
lectures on operator and systems theory and the interplay between
them. The conference was very stimulating and helped us to decide
that the time was ripe for a book on interpolation for matrix
valued functions (both rational and non-rational). When the work
started and the first partial draft of the book was ready it became
clear that the topic is vast and that the rational case by itself
with its applications is already enough material for an interesting
book. In the process of writing the book, methods for the rational
case were developed and refined. As a result we are now able to
present the rational case as an independent theory. After two years
a major part of the first draft was prepared. Then a long period of
revising the original draft and introducing recently acquired
results and methods followed. There followed a period of polishing
and of 25 chapters and the appendix commuting at various times
somewhere between Williamsburg, Blacksburg, Tel Aviv, College Park
and Amsterdam (sometimes with one or two of the authors).
This volume contains the proceedings of the Workshop on
app1ications of linear operator theory to systems and networks,
which was held at the Weizmann Institute of Science in the third
week of June, 19S3,just be fore the MTNS Conference in Beersheva.
For a 10ng time these subjects were studied indepen- dent1y by
mathematica1 ana1ysts and e1ectrica1 engineers. Never- the1ess, in
spite of the lack of communication, these two groups often
deve10ped parallel theories, though in different languages, at
different levels of genera1ity and typica11y quite different
motivations. In the last severa1 years each side has become aware
of the work of the other and there is a seeming1y ever- increasing
invo1vement of the abstract theories of factorization, extension
and interpolation of operators (and operator/matrix va1ued
functions) to the design and analysis of systems and net- works.
Moreover, the problems encountered in e1ectrica1 engineering have
genera ted new mathematica1 problems, new approaches, and usefu1
new formu1ations. The papers contained in this volume constitute a
more than representative se1ection of the presented talks and dis-
cussion at the workshop, and hopefu11y will also serve to give a
reasonably accurate picture of the problems which are under active
study today and the techniques which are used to deal with them.
The Workshop on Operator Theory and Boundary Eigenvalue Problems
was held at the Technical University, Vienna, Austria, July 27 to
30, 1993. It was the seventh workshop in the series of IWOTA
(International Workshops on Operator Theory and Applications). The
main topics at the workshop were interpolation problems and
analytic matrix functions, operator theory in spaces with
indefinite scalar products, boundary value problems for
differential and functional-differential equations and systems
theory and control. The workshop covered different aspects,
starting with abstract operator theory up to contrete applications.
The papers in these proceedings provide an accurate cross section
of the lectures presented at the workshop. This book will be of
interest to a wide group of pure and applied mathematicians.
This volume is dedicated to Harold Widom, a distinguished
mathematician and renowned expert in the area of Toeplitz,
Wiener-Hopf and pseudodifferential operators, on the occasion of
his sixtieth birthday. The book opens with biographical material
and a list of the mathematician's publications, this being followed
by two papers based on Toeplitz lectures which he delivered at Tel
Aviv University in March, 1993. The rest of the book consists of a
selection of papers containing some recent achievements in the
following areas: Szego-Widom asymptotic formulas for determinants
of finite sections of Toeplitz matrices and their generalizations,
the Fisher-Hartwig conjecture, random matrices, analysis of kernels
of Toeplitz matrices, projectional methods and eigenvalue
distribution for Toeplitz matrices, the Fredholm theory for
convolution type operators, the Nehari interpolation problem with
generalizations and applications, and Toeplitz-Hausdorff type
theorems. The book will appeal to a wide audience of pure and
applied mathematicians."
((keine o-Punkte, sondern 2 accents aigus auf dem o in
Szokefalvi, s. auch Titel ))
In August 1999, an international conference was held in Szeged,
Hungary, in honor of Bela Szokefalvi-Nagy, one of the founders and
main contributors of modern operator theory. This volume contains
some of the papers presented at the meeting, complemented by
several papers of experts who were unable to attend. These 35
refereed articles report on recent and original results in various
areas of operator theory and connected fields, many of them
strongly related to contributions of Sz.-Nagy. The scientific part
of the book is preceeded by fifty pages of biographical material,
including several photos."
These two volumes constitute texts for graduate courses in linear
operator theory. The reader is assumed to have a knowledge of both
complex analysis and the first elements of operator theory. The
texts are intended to concisely present a variety of classes of
linear operators, each with its own character, theory, techniques
and tools. For each of the classes, various differential and
integral operators motivate or illustrate the main results.
Although each class is treated seperately and the first impression
may be that of many different theories, interconnections appear
frequently and unexpectedly. The result is a beautiful, unified and
powerful theory. The classes we have chosen are representatives of
the principal important classes of operators, and we believe that
these illustrate the richness of operator theory, both in its
theoretical developments and in its applicants. Because we wanted
the books to be of reasonable size, we were selective in the
classes we chose and restricted our attention to the main features
of the corresponding theories. However, these theories have been
updated and enhanced by new developments, many of which appear here
for the first time in an operator-theory text. In the selection of
the material the taste and interest of the authors played an
important role.
This volume is devoted to the life and work of the applied
mathematician Professor Erhard Meister (1930-2001). He was a member
of the editorial boards of this book series Operator The ory:
Advances and Applications as well as of the journal Integral
Equations and Operator Theory, both published by Birkhauser (now
part of Springer-Verlag). Moreover he played a decisive role in the
foundation of these two series by helping to establish contacts
between Birkhauser and the founder and present chief editor of this
book series after his emigration from Moldavia in 1974. The volume
is divided into two parts. Part A contains reminiscences about the
life of E. Meister including a short biography and an exposition of
his professional work. Part B displays the wide range of his
scientific interests through eighteen original papers contributed
by authors with close scientific and personal relations to E.
Meister. We hope that a great part of the numerous features of his
life and work can be re-discovered from this book."
Our goal is to find Grabner bases for polynomials in four different
sets of expressions: 1 x- , (1 - x)-1 (RESOL) X, 1 x- (1 - xy)-1
(EB) X, , y-1, (1-yx)-1 y, (1_y)-1 (1-x)-1 (preNF) (EB) plus and (1
- xy)1/2 (1 - yx )1/2 (NF) (preNF) plus and Most formulas in the
theory of the Nagy-Foias operator model [NF] are polynomials in
these expressions where x = T and y = T*. Complicated polynomials
can often be simplified by applying "replacement rules". For
example, the polynomial (1 - xy)-2 - 2xy(1-xy)-2 + xy2 (1 - xy)-2
-1 simplifies to O. This can be seen by three applications of the
replacement rule (1-xy) -1 xy -t (1 - xy)-1 -1 which is true
because of the definition of (1-xy)-1. A replacement rule consists
of a left hand side (LHS) and a right hand side (RHS). The LHS will
always be a monomial. The RHS will be a polynomial whose terms are
"simpler" (in a sense to be made precise) than the LHS. An
expression is reduced by repeatedly replacing any occurrence of a
LHS by the corresponding RHS. The monomials will be well-ordered,
so the reduction procedure will terminate after finitely many
steps. Our aim is to provide a list of substitution rules for the
classes of expressions above. These rules, when implemented on a
computer, provide an efficient automatic simplification process. We
discuss and define the ordering on monomials later.
This monograph is the second volume of a graduate text book on the
modern theory of linear one-dimensional singular integral
equations. Both volumes may be regarded as unique graduate text
books. Singular integral equations attract more and more attention
since this class of equations appears in numerous applications, and
also because they form one of the few classes of equations which
can be solved explicitly. The present book is to a great extent
based upon material contained in the second part of the authors'
monograph 6] which appeared in 1973 in Russian, and in 1979 in
German translation. The present text includes a large number of
additions and complementary material, essentially changing the
character, structure and contents of the book, and making it
accessible to a wider audience. Our main subject in the first
volume was the case of closed curves and continuous coeffi cients.
Here, in the second volume, we turn to general curves and
discontinuous coefficients. We are deeply grateful to the editor
Professor G. Heinig, to the translator Dr. S. Roeh, and to the
typist Mr. G. Lillack, for their patient work. The authors
Ramat-Aviv, Ramat-Gan, May 26, 1991 11 Introduction This book is
the second volume of an introduction to the theory of linear
one-dimensional singular integral operators. The main topics of
both parts of the book are the invertibility and Fredholmness of
these operators. Special attention is paid to inversion methods."
This book is dedicated to a theory of traces and determinants on
embedded algebras of linear operators, where the trace and
determinant are extended from finite rank operators by a limit
process. The self-contained material should appeal to a wide group
of mathematicians and engineers, and is suitable for teaching.
A collection of papers on different aspects of operator theory
and complex analysis, covering the recent achievements of the
Odessa-Kharkov school, where Potapov was very active. The book
appeals to a wide group of mathematicians and engineers, and much
of the material can be used for advanced courses and seminars.
This volume is based on the proceedings of the Toeplitz Lectures
1999 and of the Workshop in Operator Theory held in March 1999 at
Tel-Aviv University and at the Weizmann Institute of Science. The
workshop was held on the occasion of the 60th birthday of Harry
Dym, and the Toeplitz lecturers were Harry Dym and Jim Rovnyak. The
papers in the volume reflect Harry's influence on the field of
operator theory and its applications through his insights, his
writings, and his personality. The volume begins with an
autobiographical sketch, followed by the list ofpublications
ofHarry Dym and the paper ofIsrael Gohberg: On Joint Work with
Harry Dym. The following paper by Jim Rovnyak: Methods of Krdn
Space Operator The- ory, is based on his Toeplitz lectures. It
gives a survey ofold and recents methods of KreIn space operator
theory along with examples from function theory, espe- cially
substitution operators on indefinite Dirichlet spaces and their
relation to coefficient problems for univalent functions, an idea
pioneered by 1. de Branges and underlying his proof of the
Bieberbach conjecture (see [9]). The remaining papers (arranged in
the alphabetical order) can be divided into the following
categories. Schur analysis and interpolation In Notes on
Interpolation in the Generalized Schur Class. I, D. Alpay, T. Con-
stantinescu, A. Dijksma, and J. Rovnyak use realization theory for
operator colli- gations in Pontryagin spaces to study interpolation
and factorization problems in generalized Schur classes.
This volume is dedicated to Tsuyoshi Ando, a foremost expert in
operator theory, matrix theory, complex analysis, and their
applications, on the occasion of his 60th birthday. The book opens
with his biography and list of publications. It contains a
selection of papers covering a broad spectrum of topics ranging
from abstract operator theory to various concrete problems and
applications. The majority of the papers deal with topics in modern
operator theory and its applications. This volume also contains
papers on interpolation and completion problems, factorization
problems and problems connected with complex analysis. The book
will appeal to a wide audience of pure and applied mathematicians.
In September 1998, during the 'International Workshop on Analysis
and Vibrat ing Systems' held in Canmore, Alberta, Canada, it was
decided by a group of participants to honour Peter Lancaster on the
occasion of his 70th birthday with a volume in the series 'Operator
Theory: Advances and Applications'. Friends and colleagues
responded enthusiastically to this proposal and within a short time
we put together the volume which is now presented to the reader.
Regarding accep tance of papers we followed the usual rules of the
journal 'Integral Equations and Operator Theory'. The papers are
dedicated to different problems in matrix and operator theory,
especially to the areas in which Peter contributed so richly. At
our request, Peter agreed to write an autobiographical paper, which
appears at the beginning of the volume. It continues with the list
of Peter's publications. We believe that this volume will pay
tribute to Peter on his outstanding achievements in different areas
of mathematics. 1. Gohberg, H. Langer P ter Lancast r *1929
Operator Theory: Advances and Applications, Vol. 130, 1- 7 (c) 2001
Birkhiiuser Verlag Basel/Switzerland My Life and Mathematics Peter
Lancaster I was born in Appleby, a small county town in the north
of England, on November 14th, 1929. I had two older brothers and
was to have one younger sister. My family moved around the north of
England as my father's work in an insurance company required."
On November 12-14, 1997 a workshop was held at the Vrije
Universiteit Amsterdam on the occasion of the sixtieth birthday
ofM. A. Kaashoek. The present volume contains the proceedings of
this workshop. The workshop was attended by 44 participants from
all over the world: partici pants came from Austria, Belgium,
Canada, Germany, Ireland, Israel, Italy, The Netherlands, South
Africa, Switzerland, Ukraine and the USA. The atmosphere at the
workshop was very warm and friendly. There where 21 plenary
lectures, and each lecture was followed by a lively discussion. The
workshop was supported by: the Vakgroep Wiskunde of the Vrije
Univer siteit, the department of Mathematics and Computer Science
of the Vrije Univer siteit, the Stichting VU Computer Science &
Mathematics Research Centre, the Thomas Stieltjes Institute for
Mathematics, and the department of Economics of the Erasmus
University Rotterdam. The organizers would like to take this
opportunity to express their gratitude for the support. Without it
the workshop would not have been so successful as it was. Table of
Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . v Photograph of M. A. Kaashoek . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Curriculum Vitae of M. A. Kaashoek . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . xv List of Publications of
M. A. Kaashoek . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . xix l. Gohberg Opening Address . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . xxxi H. Bart, A. C. M. Ran and H. I. Woerdeman Personal
Reminiscences . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . xxxv V. Adamyan and R. Mennicken
On the Separation of Certain Spectral Components of Selfadjoint
Operator Matrices . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 1 1. Introduction . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Conditions for the Separation of Spectral Components . . . . . .
. 4 3. Example . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 9 References . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . ."
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