Elementary Operator Theory

von Marat V. Markin

410 Seiten, Taschenbuch

Kurzbeschreibung des Verlags

The book is intended as a text for a one-semester graduate course in operator theory to be taught "from scratch'', not as a sequel to a functional analysis course, with the basics of the spectral theory of linear operators taking the center stage.

The book consists of
six chapters
and
appendix,
with the material flowing from the fundamentals of abstract spaces (metric, vector, normed vector, and inner product), the
Banach Fixed-Point Theorem
and its applications, such as
Picard's Existence and Uniqueness Theorem
, through the basics of linear operators, two of the three fundamental principles (the
Uniform Boundedness Principle
and the
Open Mapping Theorem
and its equivalents: the
　Inverse Mapping
and
Closed Graph Theorems
), to the elements of the spectral theory, including
Gelfand's Spectral Radius Theorem
and the
Spectral Theorem for Compact Self-Adjoint Operators
, and its applications, such as the celebrated
Lyapunov Stability Theorem
.

Conceived as a text to be used in a classroom, the book constantly calls for the student's actively mastering the knowledge of the subject matter. There are problems at the end of each chapter, starting with Chapter 2 and totaling at 150. Many important statements are given as problems and frequently referred to in the main body. There are also 432 Exercises throughout the text, including Chapter 1 and the Appendix, which require of the student to prove or verify a statement or an example, fill in certain details in a proof, or provide an intermediate step or a counterexample. They are also an inherent part of the material. More difficult problems are marked with an asterisk, many problems and exercises are supplied with "existential'' hints.　

The book is generous on Examples and contains numerous Remarks accompanying definitions, examples, and statements to discuss certain subtleties, raise questions on whether the converse assertions are true, whenever appropriate, or whether the conditions are essential.　 　　

With carefully chosen material, proper attention given to applications, and plenty of examples, problems, and exercises, this well-designed text is ideal for a one-semester Master's level graduate course in operator theory with emphasis on spectral theory for students majoring in mathematics, physics, computer science, and engineering.

Contents


Preface　　



Preliminaries

Metric Spaces

Vector Spaces, Normed Vector Spaces, and Banach Spaces

Linear Operators

Elements of Spectral Theory in a Banach Space Setting

Elements of Spectral Theory in a Hilbert Space Setting

Appendix: The Axiom of Choice and Equivalents　

Bibliography

Index

• Reihe: De Gruyter Textbook
• ISBN: 9783110600964
• Sprache: Englisch
• Erscheinungsdatum: 06.04.2020
• Genre: Mathematik/Analysis
• Verlag: De Gruyter