Linear Operators: Spectral Theory and Some Other Applications | Date: 15 April 2011, 11:34
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PREFACE
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 a.bstract algebra and analysis.
Finally, there is a fairly complete bibliography.
This work is in no sense intended to be "complete"; I give
merely those. parts of the subject which specially interest me
and which at the same time apPeared to me to fill a gap in the
literature...
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