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Author: Marat V. Markin Publisher: Walter de Gruyter GmbH & Co KG ISBN: 3110600986 Category : Mathematics Languages : en Pages : 426
Book Description
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
Author: Marat V. Markin Publisher: Walter de Gruyter GmbH & Co KG ISBN: 3110600986 Category : Mathematics Languages : en Pages : 426
Book Description
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
Author: Carlos S. Kubrusly Publisher: Springer Science & Business Media ISBN: 1475733283 Category : Mathematics Languages : en Pages : 535
Book Description
{\it Elements of Operatory Theory} is aimed at graduate students as well as a new generation of mathematicians and scientists who need to apply operator theory to their field. Written in a user-friendly, motivating style, fundamental topics are presented in a systematic fashion, i.e., set theory, algebraic structures, topological structures, Banach spaces, Hilbert spaces, culminating with the Spectral Theorem, one of the landmarks in the theory of operators on Hilbert spaces. The exposition is concept-driven and as much as possible avoids the formula-computational approach. Key features of this largely self-contained work include: * required background material to each chapter * fully rigorous proofs, over 300 of them, are specially tailored to the presentation and some are new * more than 100 examples and, in several cases, interesting counterexamples that demonstrate the frontiers of an important theorem * over 300 problems, many with hints * both problems and examples underscore further auxiliary results and extensions of the main theory; in this non-traditional framework, the reader is challenged and has a chance to prove the principal theorems anew This work is an excellent text for the classroom as well as a self-study resource for researchers. Prerequisites include an introduction to analysis and to functions of a complex variable, which most first-year graduate students in mathematics, engineering, or another formal science have already acquired. Measure theory and integration theory are required only for the last section of the final chapter.
Author: Richard V. Kadison Publisher: American Mathematical Soc. ISBN: 0821894692 Category : Mathematics Languages : en Pages : 290
Book Description
This volume is the companion volume to Fundamentals of the Theory of Operator Algebras. Volume I--Elementary Theory (Graduate Studies in Mathematics series, Volume 15). The goal of the text proper is to teach the subject and lead readers to where the vast literature--in the subject specifically and in its many applications--becomes accessible. The choice of material was made from among the fundamentals of what may be called the "classical" theory of operator algebras. This volume contains the written solutions to the exercises in the Fundamentals of the Theory of Operator Algebras. Volume I--Elementary Theory.
Author: Xavier Saint Raymond Publisher: Routledge ISBN: 1351452932 Category : Mathematics Languages : en Pages : 120
Book Description
In the 19th century, the Fourier transformation was introduced to study various problems of partial differential equations. Since 1960, this old tool has been developed into a well-organized theory called microlocal analysis that is based on the concept of the pseudo-differential operator. This book provides the fundamental knowledge non-specialists need in order to use microlocal analysis. It is strictly mathematical in the sense that it contains precise definitions, statements of theorems and complete proofs, and follows the usual method of pure mathematics. The book explains the origin of the theory (i.e., Fourier transformation), presents an elementary construcion of distribution theory, and features a careful exposition of standard pseudodifferential theory. Exercises, historical notes, and bibliographical references are included to round out this essential book for mathematics students; engineers, physicists, and mathematicians who use partial differential equations; and advanced mathematics instructors.
Author: Gerald J. Murphy Publisher: Academic Press ISBN: 0080924964 Category : Mathematics Languages : en Pages : 297
Book Description
This book constitutes a first- or second-year graduate course in operator theory. It is a field that has great importance for other areas of mathematics and physics, such as algebraic topology, differential geometry, and quantum mechanics. It assumes a basic knowledge in functional analysis but no prior acquaintance with operator theory is required.
Author: Israel Gohberg Publisher: Birkhäuser ISBN: 1461259851 Category : Mathematics Languages : en Pages : 291
Book Description
rii application of linear operators on a Hilbert space. We begin with a chapter on the geometry of Hilbert space and then proceed to the spectral theory of compact self adjoint operators; operational calculus is next presented as a nat ural outgrowth of the spectral theory. The second part of the text concentrates on Banach spaces and linear operators acting on these spaces. It includes, for example, the three 'basic principles of linear analysis and the Riesz Fredholm theory of compact operators. Both parts contain plenty of applications. All chapters deal exclusively with linear problems, except for the last chapter which is an introduction to the theory of nonlinear operators. In addition to the standard topics in functional anal ysis, we have presented relatively recent results which appear, for example, in Chapter VII. In general, in writ ing this book, the authors were strongly influenced by re cent developments in operator theory which affected the choice of topics, proofs and exercises. One of the main features of this book is the large number of new exercises chosen to expand the reader's com prehension of the material, and to train him or her in the use of it. In the beginning portion of the book we offer a large selection of computational exercises; later, the proportion of exercises dealing with theoretical questions increases. We have, however, omitted exercises after Chap ters V, VII and XII due to the specialized nature of the subject matter.
Author: Vladimir Müller Publisher: Springer Science & Business Media ISBN: 3764382651 Category : Mathematics Languages : en Pages : 444
Book Description
This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach algebras. It presents a survey of results concerning various types of spectra, both of single and n-tuples of elements. Typical examples are the one-sided spectra, the approximate point, essential, local and Taylor spectrum, and their variants. Many results appear here for the first time in a monograph.
Author: Charles W Swartz Publisher: World Scientific Publishing Company ISBN: 9813107529 Category : Mathematics Languages : en Pages : 192
Book Description
This text is an introduction to functional analysis which requires readers to have a minimal background in linear algebra and real analysis at the first-year graduate level. Prerequisite knowledge of general topology or Lebesgue integration is not required. The book explains the principles and applications of functional analysis and explores the development of the basic properties of normed linear, inner product spaces and continuous linear operators defined in these spaces. Though Lebesgue integral is not discussed, the book offers an in-depth knowledge on the numerous applications of the abstract results of functional analysis in differential and integral equations, Banach limits, harmonic analysis, summability and numerical integration. Also covered in the book are versions of the spectral theorem for compact, symmetric operators and continuous, self adjoint operators.
Author: Seymour Goldberg Publisher: Courier Corporation ISBN: 0486453316 Category : Mathematics Languages : en Pages : 212
Book Description
This volume presents a systematic treatment of the theory of unbounded linear operators in normed linear spaces with applications to differential equations. Largely self-contained, it is suitable for advanced undergraduates and graduate students, and it only requires a familiarity with metric spaces and real variable theory. After introducing the elementary theory of normed linear spaces--particularly Hilbert space, which is used throughout the book--the author develops the basic theory of unbounded linear operators with normed linear spaces assumed complete, employing operators assumed closed only when needed. Other topics include strictly singular operators; operators with closed range; perturbation theory, including some of the main theorems that are later applied to ordinary differential operators; and the Dirichlet operator, in which the author outlines the interplay between functional analysis and "hard" classical analysis in the study of elliptic partial differential equations. In addition to its readable style, this book's appeal includes numerous examples and motivations for certain definitions and proofs. Moreover, it employs simple notation, eliminating the need to refer to a list of symbols.
Author: Georgi E. Shilov Publisher: Courier Corporation ISBN: 0486318680 Category : Mathematics Languages : en Pages : 354
Book Description
Introductory text covers basic structures of mathematical analysis (linear spaces, metric spaces, normed linear spaces, etc.), differential equations, orthogonal expansions, Fourier transforms, and more. Includes problems with hints and answers. Bibliography. 1974 edition.
Author: Marat V. Markin
Author: Marat V. Markin
Author: Carlos S. Kubrusly
Author: Richard V. Kadison
Author: Xavier Saint Raymond
Author: Gerald J. Murphy
Author: Israel Gohberg
Author: Vladimir Müller
Author: Charles W Swartz
Author: Seymour Goldberg
Author: Georgi E. Shilov