The Elements of Operator Theory

verfasst von: Carlos S. Kubrusly

Verlag: Birkhäuser Boston

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This fully revised, updated, and corrected edition of The Elements of Operator Theory includes a significant expansion of problems and solutions used to illustrate the principles of operator theory. Written in a user-friendly, motivating style, it covers the fundamental topics of the field in a systematic fashion while avoiding a formula-calculation approach. The book maintains the logical and linear organization of the title’s first edition, progressing through set theory, algebraic structures, topological structures, Banach spaces, and Hilbert spaces before culminating in a discussion of the Spectral Theorem.

A self-contained textbook, The Elements of Operator Theory, Second Edition is an excellent resource for the classroom as well as a self-study reference for researchers. Prerequisites comprise an introduction to analysis and basic experience with functions of a complex variable, which most first-year graduate students in mathematics, engineering, or other formal sciences have already acquired. Measure theory and integration theory are necessary only for the last section of the final chapter.

Inhaltsverzeichnis

Frontmatter

1. Set-Theoretic Structures

Abstract

The purpose of this chapter is to present a brief review of some basic settheoretic concepts that will be needed in the sequel. By basic concepts we mean standard notation and terminology, and a few essential results that will be required in later chapters.We assume the reader is familiar with the notion of set and elements (or members, or points) of a set, as well as with the basic set operations. It is convenient to reserve certain symbols for certain sets, especially for the basic number systems. The set of all nonnegative integers will be denoted by \(\mathbb{N}_0\), the set of all positive integers (i.e., the set of all natural numbers) by \(\mathbb{N}\), and the set of all integers by \(\mathbb{Z}\). The set of all rational numbers will be denoted by \(\mathbb{Q}\), the set of all real numbers (or the real line) by \(\mathbb{R}\), and the set of all complex numbers by \(\mathbb{C}\).

Carlos S. Kubrusly

2. Algebraic Structures

Abstract

The main algebraic structure involved with the subject of this book is that of a “linear space” (or “vector space”). A linear space is a set endowed with an extra structure in addition to its set-theoretic structure (i.e., an extra structure that goes beyond the notions of inclusion, union, complement, function, and ordering, for instance). Roughly speaking, linear spaces are sets where two operations, called “addition” and “scalar multiplication”, are properly defined so that we can refer to the “sum” of two points in a linear space, as well as to the “product” of a point in it by a “scalar”. Although the reader is supposed to have already had a contact with linear algebra and, in particular, with “finite-dimensional vector spaces”, we shall proceed from the very beginning. Our approach avoids the parochially “finite-dimensional” constructions (whenever this is possible), and focuses either on general results that do not depend on the “dimensionality” of the linear space, or on abstract “infinitedimensional” linear spaces.

Carlos S. Kubrusly

3. Topological Structures

Abstract

The basic concept behind the subject of point-set topology is the notion of “closeness” between two points in a set X. In order to get a numerical gauge of how close together two points in X may be, we shall provide an extra structure to X, viz., a topological structure, that again goes beyond its purely settheoretic structure. For most of our purposes the notion of closeness associated with a metric will be sufficient, and this leads to the concept of “metric space”: a set upon which a “metric” is defined. The metric-space structure that a set acquires when a metric is defined on it is a special kind of topological structure. Metric spaces comprise the kernel of this chapter, but general topological spaces are also introduced.

Carlos S. Kubrusly

4. Banach Spaces

Abstract

Our purpose now is to put algebra and topology to work together. For instance, from algebra we get the notion of finite sums (either ordinary or direct sums of vectors, linear manifolds, or linear transformations), and from topology the notion of convergent sequences. If algebraic and topological structures are suitably laid on the same underlying set, then we may consider the concept of infinite sums and convergent series. More importantly, as continuity plays a central role in the theory of topological spaces, and linear transformation plays a central role in the theory of linear spaces, when algebra and topology are properly combined they yield the concept of continuous linear transformation; the very central theme of this book.

Carlos S. Kubrusly

5. Hilbert Spaces

Abstract

What is missing? The algebraic structure of a normed space allowed us to operate with vectors (addition and scalar multiplication), and its topological structure (the one endowed by the norm) gave us a notion of closeness (by means of the metric generated by the norm), which interacts harmoniously with the algebraic operations. In particular, it provided the notion of length of a vector. So what is missing if algebra and topology have already been properly laid on the same underlying set? A full geometric structure is still missing. Algebra and topology are not enough to extend to abstract spaces the geometric concept of relative direction (or angle) between vectors that is familiar in Euclidean geometry. The keyword here is orthogonality, a concept that emerges when we equip a linear space with an inner product. This supplies a tremendously rich structure that leads to remarkable simplifications.

Carlos S. Kubrusly

6. The Spectral Theorem

Abstract

The Spectral Theorem is a landmark in the theory of operators on Hilbert space, providing a full statement about the nature and structure of normal operators. Normal operators play a central role in operator theory; they will be defined in Section 6.1 below. It is customary to say that the Spectral Theorem can be applied to answer essentially all questions on normal operators. This indeed is the case as far as “essentially all” means “almost all” or “all the principal”: there exist open questions on normal operators. First we consider the class of normal operators and its relatives (predecessors and successors). Next, the notion of spectrum of an operator acting on a complex Banach space is introduced. The Spectral Theorem for compact normal operators is fully investigated, yielding the concept of diagonalization. The Spectral Theorem for plain normal operators needs measure theory. We would not dare to relegate measure theory to an appendix just to support a proper proof of the Spectral Theorem for plain normal operators. Instead we assume just once, in the very last section of this book, that the reader has some familiarity with measure theory, just enough to grasp the statement of the Spectral Theorem for plain normal operators after having proved it for compact normal operators.

Carlos S. Kubrusly

Backmatter

Titel: The Elements of Operator Theory
verfasst von: Carlos S. Kubrusly
Verlag: Birkhäuser Boston
Electronic ISBN: 978-0-8176-4998-2
Print ISBN: 978-0-8176-4997-5
DOI: https://doi.org/10.1007/978-0-8176-4998-2

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Set-Theoretic Structures

2. Algebraic Structures

3. Topological Structures

4. Banach Spaces

5. Hilbert Spaces

6. The Spectral Theorem

Backmatter

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