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This volume is dedicated to Leonid Lerer on the occasion of his
seventieth birthday. The main part presents recent results in
Lerer's research area of interest, which includes Toeplitz,
Toeplitz plus Hankel, and Wiener-Hopf operators, Bezout equations,
inertia type results, matrix polynomials, and related areas in
operator and matrix theory. Biographical material and Lerer's list
of publications complete the volume.
This volume is dedicated to Leonid Lerer on the occasion of his
seventieth birthday. The main part presents recent results in
Lerer's research area of interest, which includes Toeplitz,
Toeplitz plus Hankel, and Wiener-Hopf operators, Bezout equations,
inertia type results, matrix polynomials, and related areas in
operator and matrix theory. Biographical material and Lerer's list
of publications complete the volume.
Many developments on the cutting edge of research in operator
theory and its applications are reflected in this collection of
original and review articles. Particular emphasis lies on
highlighting the interplay between operator theory and applications
from other areas, such as multi-dimensional systems and function
theory of several complex variables, distributed parameter systems
and control theory, mathematical physics, wavelets, and numerical
analysis.
This is a collection of original and review articles on recent
advances and new directions in a multifaceted and interconnected
area of mathematics and its applications. It encompasses many
topics in theoretical developments in operator theory and its
diverse applications in applied mathematics, physics, engineering,
and other disciplines. The purpose is to bring in one volume many
important original results of cutting edge research as well as
authoritative review of recent achievements, challenges, and future
directions in the area of operator theory and its applications. The
intended audience are mathematicians, physicists, electrical
engineers in academia and industry, researchers and graduate
students, that use methods of operator theory and related fields of
mathematics, such as matrix theory, functional analysis,
differential and difference equations, in their work.
Thefollowing topics ofmathematical analysishavebeen developed in
the last?fty years:
thetheoryoflinearcanonicaldi?erentialequationswithperiodicHamilto-
ans, the theory of matrix polynomials with selfadjoint coe?cients,
linear di?er- tial and di?erence equations of higher order with
selfadjoint constant coe?cients, andalgebraicRiccati equations.All
of these theories, and others, arebased on r- atively recent
results of linear algebra in spaces with an inde?nite inner
product, i.e., linear algebra in which the usual positive de?nite
inner product is replaced by an inde?nite one. More concisely, we
call this subject inde?nite linear algebra. This book has the
structureof a graduatetext in which chaptersof advanced linear
algebra form the core. The development of our topics follows the
lines of a usual linear algebra course. However, chapters giving
comprehensive treatments of di?erential and di?erence equations,
matrix polynomials and Riccati equations are interwoven as the
necessary techniques are developed. The main source of material is
our earlier monograph in this ?eld: Matrices and Inde?nite Scalar
Products, 40]. The present book di?ers in objectives and
material.Somechaptershavebeenexcluded, othershavebeenadded,
andexercises have been added to all chapters. An appendix is also
included. This may serve as a summary and refresher on standard
results as well as a source for some less familiar material from
linear algebra with a de?nite inner product. The theory developed
here has become an essential part of linear algebra. This, together
with the many signi?cant areas of application, and the accessible
style, make this book useful for engineers, scientists and
mathematicians al
Many developments on the cutting edge of research in operator
theory and its applications are reflected in this collection of
original and review articles. Particular emphasis lies on
highlighting the interplay between operator theory and applications
from other areas, such as multi-dimensional systems and function
theory of several complex variables, distributed parameter systems
and control theory, mathematical physics, wavelets, and numerical
analysis.
This unique book addresses advanced linear algebra from a
perspective in which invariant subspaces are the central notion and
main tool. It contains comprehensive coverage of geometrical,
algebraic, topological, and analytic properties of invariant
subspaces. The text lays clear mathematical foundations for linear
systems theory and contains a thorough treatment of analytic
perturbation theory for matrix functions.
Quaternions are a number system that has become increasingly
useful for representing the rotations of objects in
three-dimensional space and has important applications in
theoretical and applied mathematics, physics, computer science, and
engineering. This is the first book to provide a systematic,
accessible, and self-contained exposition of quaternion linear
algebra. It features previously unpublished research results with
complete proofs and many open problems at various levels, as well
as more than 200 exercises to facilitate use by students and
instructors. Applications presented in the book include numerical
ranges, invariant semidefinite subspaces, differential equations
with symmetries, and matrix equations.
Designed for researchers and students across a variety of
disciplines, the book can be read by anyone with a background in
linear algebra, rudimentary complex analysis, and some
multivariable calculus. Instructors will find it useful as a
complementary text for undergraduate linear algebra courses or as a
basis for a graduate course in linear algebra. The open problems
can serve as research projects for undergraduates, topics for
graduate students, or problems to be tackled by professional
research mathematicians. The book is also an invaluable reference
tool for researchers in fields where techniques based on quaternion
analysis are used.
New versions are developed of an abstract scheme, which are
designed to provide a framework for solving a variety of extension
problems. The abstract scheme is commonly known as the band method.
The main feature of the new versions is that they express directly
the conditions for existence of positive band extensions in terms
of abstract factorizations (with certain additional properties).
The results allow us to prove, among other things, that the band
extension is continuous in an appropriate sense. Using the new
versions of the abstract band method, we solve the positive
extension problem for almost periodic matrix functions of several
real variables with Fourier coefficients indexed in a given
additive subgroup of the space of variables.This generality allows
us to treat simultaneously many particular cases, for example the
case of functions periodic in some variables and almost periodic in
others. Necessary and sufficient conditions are given for the
existence of positive extensions in terms of Toeplitz operators on
Besikovitch spaces. Furthermore, when a solution exists a special
extension (the band extension) is constructed which enjoys a
maximum entropy property.A linear fractional parameterization of
the set of all extensions is also provided. We interpret the
obtained results (in the periodic case) in terms of existence of a
multivariate autoregressive moving averages (ARMA) process with
given autocorrelation coefficients, and identify its maximal
prediction error. Another application concerns the solution of the
positive extension problem in the context of Wiener algebra of
infinite operator matrices. It includes the identification of the
maximum entropy extension and a description of all positive
extensions via a linear fractional formula. In the periodic case it
solves a linear estimation problem for cyclostationary stochastic
processes.
This book provides a careful treatment of the theory of algebraic
Riccati equations. It consists of four parts: the first part is a
comprehensive account of necessary background material in matrix
theory including careful accounts of recent developments involving
indefinite scalar products and rational matrix functions. The
second and third parts form the core of the book and concern the
solutions of algebraic Riccati equations arising from continuous
and discrete systems. The geometric theory and iterative analysis
are both developed in detail. The last part of the book is an
exciting collection of eight problem areas in which algebraic
Riccati equations play a crucial role. These applications range
from introductions to the classical linear quadratic regulator
problems and the discrete Kalman filter to modern developments in
HD*W*w control and total least squares methods.
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