Cooke R.C. Linear Operators: Spectral Theory and Some Other Applications

Macmillan, 1953. - 466 pages.THE consideration of linear operators follows in a natural way from a study of infinite matrices, which form an important particular case when the matrices are associative. This book is, however, complete in itself, in that it is not essential to know more than the most elementary properties of infinite matrices; where such knowledge is required, references to my Infinite Matrices and Sequence Spaces (Macmillan, 1950) are given in the text.One of the chief applications of linear operators to date is to Quantum Mechanics-in particular, to proofs of the spectral resolution theorem for unbounded self-adjoint (hypermaximal)
operators. No less than eleven suoh proofs have been published up to the present time (1951), and four of these are given in the present text, with summaries of the others. The proof of Lengyel and Stone for bounded operators is also given, since several of the proofs in the unbounded case assume the bounded case as already established. Each of these eleven proofs has its own special features of interest.Chapter 1 in the present work is concerned with Hilbert functional space and abstract Hilbert space, this being the medium in which the above eleven proofs are worked out for Hilbert vector (or sequence) space, see my Infinite Matrices and Sequence Spaces, Chapter 9.
Chapter 2 differs from the remaining chapters of the book; whereas the latter are entirely pure mathematical, an attempt is made in Chapter 2 to show the connection of infinite matrices
and linear operators in general with the physicaJ background of Quantum Mechanics, so as to explain the reasons for the otherwise seemingly artificial problems which are considered in Chapters 3, 4, and
5. In a single chapter this is necessarily given in an exceedingly sketchy manner, mainly in the form of an historioal summary. The pure mathematician who is repelled by this chapter may omit it without detriment to his understanding of the rest of the book; although a perusal of at least §§ 2.6 and 2.7 would be an advantage. Chapter 3 gives an account of linear operators in Hilbert space, and of the deficiency indices, and provides the tools
required for Chapter 4, which deals with the first proof given of the general spectral resolution theorem-that of von Neumann, and for Chapter 5, which gives the other proofs of the theorem referred to above. Chapter 6 is devoted to a totally different application of linear operators, namely to matrix spaces and rings; it comprises recent work by Kothe and Toeplitz, Weber, and Allen. There is no account of these infinite matrix rings given in any other book. Chapter 7 gives an introduction to Banach algebras (normed rings), with applications to Wiener's theorems on absolutely convergent trigonometric series and integrals, to Bochner's theorem on positive-definite functions (the usual proof of which is given in § 6.4), and Wiener's theorem on the closure of translations. This topological algebra forms a very interesting link between abstract algebra and analysis.