Joel H. Shapiro – författare

The study of composition operators links some of the most basic questions you can ask about linear operators with beautiful classical results from analytic-function theory. The process invests old theorems with new mean­ ings, and bestows upon functional analysis an intriguing class of concrete linear operators. Best of all, the subject can be appreciated by anyone with an interest in function theory or functional analysis, and a background roughly equivalent to the following twelve chapters of Rudin's textbook Real and Complex Analysis [Rdn '87]: Chapters 1-7 (measure and integra­ tion, LP spaces, basic Hilbert and Banach space theory), and 10-14 (basic function theory through the Riemann Mapping Theorem). In this book I introduce the reader to both the theory of composition operators, and the classical results that form its infrastructure. I develop the subject in a way that emphasizes its geometric content, staying as much as possible within the prerequisites set out in the twelve fundamental chapters of Rudin's book. Although much of the material on operators is quite recent, this book is not intended to be an exhaustive survey. It is, quite simply, an invitation to join in the fun. The story goes something like this.

The study of composition operators links some of the most basic questions you can ask about linear operators with beautiful classical results from analytic-function theory. The process invests old theorems with new mean­ ings, and bestows upon functional analysis an intriguing class of concrete linear operators. Best of all, the subject can be appreciated by anyone with an interest in function theory or functional analysis, and a background roughly equivalent to the following twelve chapters of Rudin''s textbook Real and Complex Analysis [Rdn ''87]: Chapters 1-7 (measure and integra­ tion, LP spaces, basic Hilbert and Banach space theory), and 10-14 (basic function theory through the Riemann Mapping Theorem). In this book I introduce the reader to both the theory of composition operators, and the classical results that form its infrastructure. I develop the subject in a way that emphasizes its geometric content, staying as much as possible within the prerequisites set out in the twelve fundamental chapters of Rudin''s book. Although much of the material on operators is quite recent, this book is not intended to be an exhaustive survey. It is, quite simply, an invitation to join in the fun. The story goes something like this.

This book introduces functional analysis to undergraduate mathematics students who possess a basic background in analysis and linear algebra. By studying how the Volterra operator acts on vector spaces of continuous functions, its readers will sharpen their skills, reinterpret what they already know, and learn fundamental Banach-space techniques--all in the pursuit of two celebrated results: the Titchmarsh Convolution Theorem and the Volterra Invariant Subspace Theorem. Exercises throughout the text enhance the material and facilitate interactive study.

This book aims to investigate how Riesz and Fredholm operator theories extend and adapt within the framework of topological vector spaces. At its core, this book seeks to generalize classical operator theory, traditionally developed in Banach and Hilbert spaces, to more inclusive topological settings. It offers a unified and comprehensive approach to understanding Riesz and Fredholm theories beyond the confines of local convexity and in the absence of an abundance of continuous linear functionals (which are the prevalent assumptions in existing literature). The exposition begins with foundational concepts in topology, including open and closed sets, neighborhoods, and compactness. It then introduces metrics, norms, and completeness, followed by a treatment of vector spaces, subspaces, and linear topological structures. The discussion progresses to inner products, orthogonality, and Hilbert space geometry. The heart of the book is the development of Riesz theory in topological vector spaces, beginning with compact operators, eigenvalues, and eigensubspaces, and advancing to spectral theory. This sets the stage for constructing a Fredholm theory in general topological contexts, requiring new definitions and a thorough exploration of the Fredholm index, its stability, and its applications. The book also includes a wide range of exercises (from foundational proofs to advanced problems) designed to reinforce the content covered and to provide additional insights.

This book aims to investigate how Riesz and Fredholm operator theories extend and adapt within the framework of topological vector spaces. At its core, this book seeks to generalize classical operator theory, traditionally developed in Banach and Hilbert spaces, to more inclusive topological settings. It offers a unified and comprehensive approach to understanding Riesz and Fredholm theories beyond the confines of local convexity and in the absence of an abundance of continuous linear functionals (which are the prevalent assumptions in existing literature). The exposition begins with foundational concepts in topology, including open and closed sets, neighborhoods, and compactness. It then introduces metrics, norms, and completeness, followed by a treatment of vector spaces, subspaces, and linear topological structures. The discussion progresses to inner products, orthogonality, and Hilbert space geometry. The heart of the book is the development of Riesz theory in topological vector spaces, beginning with compact operators, eigenvalues, and eigensubspaces, and advancing to spectral theory. This sets the stage for constructing a Fredholm theory in general topological contexts, requiring new definitions and a thorough exploration of the Fredholm index, its stability, and its applications. The book also includes a wide range of exercises (from foundational proofs to advanced problems) designed to reinforce the content covered and to provide additional insights.

This text provides an introduction to some of the best-known fixed-point theorems, with an emphasis on their interactions with topics in analysis.  The level of exposition increases gradually throughout the book, building from a basic requirement of undergraduate proficiency to graduate-level sophistication. Appendices provide an introduction to (or refresher on) some of the prerequisite material and exercises are integrated into the text, contributing to the volume’s ability to be used as a self-contained text. Readers will find the presentation especially useful for independent study or as a supplement to a graduate course in fixed-point theory.The material is split into four parts: the first introduces the Banach Contraction-Mapping Principle and the Brouwer Fixed-Point Theorem, along with a selection of interesting applications; the second focuses on Brouwer’s theorem and its application to John Nash’s work; the third applies Brouwer’s theorem to spaces of infinite dimension; and the fourth rests on the work of Markov, Kakutani, and Ryll–Nardzewski surrounding fixed points for families of affine maps.

This text provides an introduction to some of the best-known fixed-point theorems, with an emphasis on their interactions with topics in analysis.  The level of exposition increases gradually throughout the book, building from a basic requirement of undergraduate proficiency to graduate-level sophistication. Appendices provide an introduction to (or refresher on) some of the prerequisite material and exercises are integrated into the text, contributing to the volume’s ability to be used as a self-contained text. Readers will find the presentation especially useful for independent study or as a supplement to a graduate course in fixed-point theory.

The material is split into four parts: the first introduces the Banach Contraction-Mapping Principle and the Brouwer Fixed-Point Theorem, along with a selection of interesting applications; the second focuses on Brouwer’s theorem and its application to John Nash’s work; the third applies Brouwer’s theorem to spaces of infinite dimension; and the fourth rests on the work of Markov, Kakutani, and Ryll–Nardzewski surrounding fixed points for families of affine maps.