The Cauchy transform of a measure on the circle is a subject of both classical and current interest with a sizable literature. This book is a thorough, well-documented, and readable survey of this literature and includes full proofs of the main results of the subject. This book also covers more recent perturbation theory as covered by Clark, Poltoratski, and Aleksandrov and contains an in-depth treatment of Clark measures.
Proofs are presented very carefully, so that the whole material becomes accessible to any graduate student interested in analysis. Experienced researchers will also be thankful to the authors for having presented them this material for the first time. We recommend this survey to any person interested in function theory and operator theory on the disk.
-- Zentralblatt MATH
The book is user-friendly it is clearly written, at an agreeable pace. Many of the deepest results of the theory are afforded complete treatments. Scattered throughout are mentions of related results, with references to the literature. The authors are thorough and meticulous in their citation of references.
-- Mathematical Reviews
...contains a wealth of interesting results, presented in an attractive and readable style...
-- Bulletin of the LMS
Combining both classical and recent result, the book presents a great interest for students, teachers and researchers interested mainly in functional analysis methods in complex analysis. The topics are presented in an elegant manner, with many comments, detours and historical references. The result is a fine book that deserves to be on the bookshelf of each analyst.
-- Studia Universitatis Babes-Bolyai, Mathematica
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 \(H^2\) 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 \(1 < p < \infty\) was done in
a 1970 paper of R. Douglas, H. S. Shapiro, and A. Shields, and the case
\(0 < p \le 1\) 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 \(0 < p < 1\)
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.
Advanced graduate students with a background in basic functional analysis, complex analysis and the basics of the theory of Hardy spaces; professional mathematicians interested in operator theory and functional analysis.
The book has been carefully written and contains a wealth of information … It will probably appeal most to those with an interest in the interplay between operator theory and modern function theory.
-- Bulletin of the LMS
This volume contains the Proceedings of the Conference on Completeness
Problems, Carleson Measures, and Spaces of Analytic Functions, held
from June 29–July 3, 2015, at the Institut Mittag-Leffler,
Djursholm, Sweden.
The conference brought together experienced researchers and promising
young mathematicians from many countries to discuss recent progress
made in function theory, model spaces, completeness problems, and
Carleson measures.
This volume contains articles covering cutting-edge research
questions, as well as longer survey papers and a report on the problem
session that contains a collection of attractive open problems in
complex and harmonic analysis.
Graduate students and research mathematicians interested in operator theory and complex analysis.
This volume contains the proceedings of the CRM Workshop on Invariant
Subspaces of the Shift Operator, held August 26–30, 2013, at the
Centre de Recherches Mathématiques, Université de Montréal,
Montréal, Quebec, Canada.
The main theme of this volume is the invariant subspaces of the
shift operator (or its adjoint) on certain function spaces, in
particular, the Hardy space, Dirichlet space, and de
Branges–Rovnyak spaces.
These spaces, and the action of the shift operator on them, have
turned out to be a precious tool in various questions in analysis such
as function theory (Bieberbach conjecture, rigid functions,
Schwarz–Pick inequalities), operator theory (invariant subspace
problem, composition operator), and systems and control theory.
Of particular interest is the Dirichlet space, which is one of the
classical Hilbert spaces of holomorphic functions on the unit
disk. From many points of view, the Dirichlet space is an interesting
and challenging example of a function space. Though much is known
about it, several important open problems remain, most notably the
characterization of its zero sets and of its shift-invariant
subspaces.
Graduate students and research mathematicians interested in operator theory and function spaces.
The theory of generalized analytic continuation studies
continuations of meromorphic functions in situations where traditional
theory says there is a natural boundary. This broader theory
touches on a remarkable array of topics in classical analysis, as
described in the book. This book addresses the following questions:
(1) When can we say, in some reasonable way, that component functions
of a meromorphic function on a disconnected domain, are
“continuations” of each other? (2) What role do such
“continuations” play in certain aspects of approximation
theory and operator theory? The authors use the strong analogy with
the summability of divergent series to motivate the subject. In this
vein, for instance, theorems can be described as being
“Abelian” or “Tauberian”. The introductory
overview carefully explains the history and context of the
theory.
The authors begin with a review of the works of Poincaré,
Borel, Wolff, Walsh, and Gončar, on continuation properties of
“Borel series” and other meromorphic functions that are
limits of rapidly convergent sequences of rational functions. They
then move on to the work of Tumarkin, who looked at the continuation
properties of functions in the classical Hardy space of the disk in
terms of the concept of “pseudocontinuation”. Tumarkin's
work was seen in a different light by Douglas, Shapiro, and Shields in
their discovery of a characterization of the cyclic vectors for the
backward shift operator on the Hardy space. The authors cover this
important concept of “pseudocontinuation” quite thoroughly
since it appears in many areas of analysis. They also add a new and
previously unpublished method of “continuation” to the
list, based on formal multiplication of trigonometric series, which
can be used to examine the backward shift operator on many spaces of
analytic functions. The book attempts to unify the various types of
“continuations” and suggests some interesting open
questions.
Graduate students and research mathematicians interested in functions of a complex variable, approximation theory, and operator theory.
Interesting and well written book … an extensive and useful bibliography …
-- Zentralblatt MATH
Interesting and inspiring small book … can be recommended as a source of many interesting and well-motivated open problems.
-- Mathematical Reviews