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## Über dieses Buch

Operator theory is a diverse area of mathematics which derives its impetus and motivation from several sources. It began with the study of integral equations and now includes the study of operators and collections of operators arising in various branches of physics and mechanics. The intention of this book is to discuss certain advanced topics in operator theory and to provide the necessary background for them assuming only the standard senior-first year graduate courses in general topology, measure theory, and algebra. At the end of each chapter there are source notes which suggest additional reading along with giving some comments on who proved what and when. In addition, following each chapter is a large number of problems of varying difficulty. This new edition will appeal to a new generation of students seeking an introduction to operator theory.

## Inhaltsverzeichnis

### Chapter 1. Banach Spaces

Abstract
1.1 We begin by introducing the most representative example of a Banach space. LetXbe a compact Hausdorff space and letC(X)denote the set of continuous complex-valued functions onX.For fiand f2 inC(X)and X a complex number, we define:
(1)
(f 1+f 2)(x)=f 1(x)+f 2(x)

(2)
f 1)(x)=λf 1(x); and

(3)
(f 1 f 2)(x)=f 1(x)f 2(x)

Ronald G. Douglas

### Chapter 2. Banach Algebras

Abstract
In Chapter 1 we showed thatC(X)is a Banach space and that every Banach space is, in fact, isomorphic to a subspace of someC(X).In addition to being a linear spaceC(X)is also an algebra and multiplication is continuous in the norm topology. In this chapter we studyC(X)as a Banach algebra and show thatC(X)is a “universal” commutative Banach algebra in a sense which we will later make precise. We shall indicate the usefulness and power of this result in some examples.
Ronald G. Douglas

### Chapter 3. Geometry of Hilbert Space

Abstract
The notion of Banach space abstracts many of the important properties of finite-dimensional linear spaces. The geometry of a Banach space can, however, be quite different from that of Euclidean n-space; for example, the unit ball of a Banach space may have corners, and closed convex sets need not possess a unique vector of smallest norm. The most important geometrical property absent in general Banach spaces is a notion of perpendicularity or orthogonality.
Ronald G. Douglas

### Chapter 4. Operators on Hilbert Space and C*-Algebras

Abstract
Most of linear algebra involves the study of transformations between linear spaces which preserve the linear structure, that is, linear transformations. Such is also the case in the study of Hilbert spaces. In the remainder of the book we shall be mainly concerned with bounded linear transformations acting on Hilbert spaces. Despite the importance of certain classes of unbounded linear transformations, we consider them only in the problems.
Ronald G. Douglas

### Chapter 5. Compact Operators, Fredholm Operators, and Index Theory

Abstract
In the preceding chapter we studied operators on Hilbert space and obtained, in particular, the spectral theorem for normal operators. As we indicated this result can be viewed as the appropriate generalization to infinite-dimensional spaces of the diagonalizability of matrices on finite-dimensional spaces. There is another class of operators which are a generalization in a topological sense of operators on a finite-dimensional space. In this chapter we study these operators and a certain related class. The organization of our study is somewhat unorthodox and is arranged so that the main results are obtained as quickly as possible. We first introduce the class of compact operators and show that this class coincides with the norm closure of the finite rank operators. After that we give some concrete examples of compact operators and then proceed to introduce the notion of a Fredholm operator. We begin with a definition.
Ronald G. Douglas

### Chapter 6. The Hardy Spaces

Abstract
In this chapter we study various properties of the spacesH 1 H 2 andH in preparation for our study of Toeplitz operators in the following chapter. Due to the availability of several excellent accounts of this subject (see Notes), we do not attempt a comprehensive treatment and proceed in the main using the techniques which we have already introduced.
Ronald G. Douglas

### Chapter 7. Toeplitz Operators

Abstract
Despite considerable effort there are few classes of operators on Hilbert space which one can declare are fully understood. Except for the self-adjoint operators and a few other examples, very little is known about the detailed structure of any class of operators. In fact, in most cases even the appropriate questions are not clear. In this chapter we study a class of operators about which much is known and even more remains to be known. Although the results we obtain would seem to fully justify their study, the occurrence of this class of operators in other areas of mathematics suggests they play a larger role in operator theory than would at first be obvious.
Ronald G. Douglas

### Backmatter

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