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This monograph offers an introduction to finite Blaschke products
and their connections to complex analysis, linear algebra, operator
theory, matrix analysis, and other fields. Old favorites such as
the Caratheodory approximation and the Pick interpolation theorems
are featured, as are many topics that have never received a modern
treatment, such as the Bohr radius and Ritt's theorem on
decomposability. Deep connections to hyperbolic geometry are
explored, as are the mapping properties, zeros, residues, and
critical points of finite Blaschke products. In addition, model
spaces, rational functions with real boundary values, spectral
mapping properties of the numerical range, and the Darlington
synthesis problem from electrical engineering are also covered.
Topics are carefully discussed, and numerous examples and
illustrations highlight crucial ideas. While thorough explanations
allow the reader to appreciate the beauty of the subject, relevant
exercises following each chapter improve technical fluency with the
material. With much of the material previously scattered throughout
mathematical history, this book presents a cohesive, comprehensive
and modern exposition accessible to undergraduate students,
graduate students, and researchers who have familiarity with
complex analysis.
If H is a Hilbert space and T : H ? H is a continous linear
operator, a natural question to ask is: What are the closed
subspaces M of H for which T M ? M? Of course the famous invariant
subspace problem asks whether or not T has any non-trivial
invariant subspaces. This monograph is part of a long line of study
of the invariant subspaces of the operator T = M (multiplication by
the independent variable z, i. e. , M f = zf )on a z z Hilbert
space of analytic functions on a bounded domain G in C. The
characterization of these M -invariant subspaces is particularly
interesting since it entails both the properties z of the functions
inside the domain G, their zero sets for example, as well as the
behavior of the functions near the boundary of G. The operator M is
not only interesting in its z own right but often serves as a model
operator for certain classes of linear operators. By this we mean
that given an operator T on H with certain properties (certain
subnormal operators or two-isometric operators with the right
spectral properties, etc. ), there is a Hilbert space of analytic
functions on a domain G for which T is unitarity equivalent to M .
The classical $\ell^{p}$ sequence spaces have been a mainstay in
Banach spaces. This book reviews some of the foundational results
in this area (the basic inequalities, duality, convexity, geometry)
as well as connects them to the function theory (boundary growth
conditions, zero sets, extremal functions, multipliers, operator
theory) of the associated spaces $\ell^{p}_{A}$ of analytic
functions whose Taylor coefficients belong to $\ell^p$. Relations
between the Banach space $\ell^p$ and its associated function space
are uncovered using tools from Banach space geometry, including
Birkhoff-James orthogonality and the resulting Pythagorean
inequalities. The authors survey the literature on all of this
material, including a discussion of the multipliers of
$\ell^{p}_{A}$ and a discussion of the Wiener algebra
$\ell^{1}_{A}$. Except for some basic measure theory, functional
analysis, and complex analysis, which the reader is expected to
know, the material in this book is self-contained and detailed
proofs of nearly all the results are given. Each chapter concludes
with some end notes that give proper references, historical
background, and avenues for further exploration.
Aimed at graduate students, this textbook provides an accessible
and comprehensive introduction to operator theory. Rather than
discuss the subject in the abstract, this textbook covers the
subject through twenty examples of a wide variety of operators,
discussing the norm, spectrum, commutant, invariant subspaces, and
interesting properties of each operator. The text is supplemented
by over 600 end-of-chapter exercises, designed to help the reader
master the topics covered in the chapter, as well as providing an
opportunity to further explore the vast operator theory literature.
Each chapter also contains well-researched historical facts which
place each chapter within the broader context of the development of
the field as a whole.
Aimed at graduate students, this textbook provides an accessible
and comprehensive introduction to operator theory. Rather than
discuss the subject in the abstract, this textbook covers the
subject through twenty examples of a wide variety of operators,
discussing the norm, spectrum, commutant, invariant subspaces, and
interesting properties of each operator. The text is supplemented
by over 600 end-of-chapter exercises, designed to help the reader
master the topics covered in the chapter, as well as providing an
opportunity to further explore the vast operator theory literature.
Each chapter also contains well-researched historical facts which
place each chapter within the broader context of the development of
the field as a whole.
The study of model spaces, the closed invariant subspaces of the
backward shift operator, is a vast area of research with
connections to complex analysis, operator theory and functional
analysis. This self-contained text is the ideal introduction for
newcomers to the field. It sets out the basic ideas and quickly
takes the reader through the history of the subject before ending
up at the frontier of mathematical analysis. Open questions point
to potential areas of future research, offering plenty of
inspiration to graduate students wishing to advance further.
Shift operators on Hilbert spaces of analytic functions play an
important role in the study of bounded linear operators on Hilbert
spaces since they often serve as models for various classes of
linear operators. For example, 'parts' of direct sums of the
backward shift operator on the classical Hardy space H2 model
certain types of contraction operators and potentially have
connections to understanding the invariant subspaces of a general
linear operator. This book is a thorough treatment of the
characterization of the backward shift invariant subspaces of the
well-known Hardy spaces H{p}. The characterization of the backward
shift invariant subspaces of H{p} for 1 infinity was done in a 1970
paper of R. Douglas, H. S. Shapiro, and A. Shields, and the case
0p\le 1 was done in a 1979 paper of A. B. Aleksandrov which is not
well known in the West.This material is pulled together in this
single volume and includes all the necessary background material
needed to understand (especially for the 0 1 case) the proofs of
these results. Several proofs of the Douglas-Shapiro-Shields result
are provided so readers can get acquainted with different operator
theory and theory techniques: applications of these proofs are also
provided for understanding the backward shift operator on various
other spaces of analytic functions. The results are thoroughly
examined. Other features of the volume include a description of
applications to the spectral properties of the backward shift
operator and a treatment of some general real-variable techniques
that are not taught in standard graduate seminars. The book
includes references to works by Duren, Garnett, and Stein for
proofs and a bibliography for further exploration in the areas of
operator theory and functional analysis.
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