
Author(s)  Samoilenko, Y.S. 

Title  Spectral Theory of Families of SelfAdjoint Operators 
Publisher  Kluwer Academic Publishers 
Year of publication  1991 
Reviewed by  S. Cobzas 
This book is dedicated to the construction of mathematical models for a number of physical systems in the language of families of commuting selfadjoint operators. The spectral theory of finitely many commuting selfadjoint operators is treated in the book of M.Sh. Birman and M.Z. Solomyak, The spectral theory of selfadjoint operators in Hilbert spaces, Leningrad University Press, Leningrad1980 (in Russian), and the applications of selfadjoint operators in modern mathematical physics is presented largely in the four volumes of the book of M. Reed and B. Simon, Methods of modern mathematical physics, Academic Press, from 1972 to 1978. The spectral theory for arbitrary systems of commuting selfadjoint operators was treated by Yu. A. Berezanskii, Selfadjoint operators in spaces of functions of infinitely many variables, Naukova Dumka, Kiev 1978, English translation, Amer. Math. Soc., Providence R.I., 1986. The author concentrates in this book on a special case of such systems, namely countable collections of commuting selfadjoint operators and their joint spectral properties. Such collections have a lot of additional properties which yield a simplification of the formulations of general theorems as well as a number of new assertions, similar to their finite dimensional analogues. There exists a comparable situation in the study of random sequence, because one can consider a commuting family of selfadjoint operators as a "nonrealized" random field. The correspondence between problems in the theory of random processes and the theory of infinite families of commuting operators is treated in the fourth part of the book.
The theory of countable families of commuting selfadjoint operators is also connected with the theory of unitary representations of inductive limits of commutative locally compact simply connected Lie groups, a topic treated in the second chapter of the book. These theories are then applied in Chapter 3 to the study of differential operators with constant coefficients on functions of countably many variables.
The book is divided into four parts and twelve chapters: Part I, Families of commuting normal operators; Part II. Inductive limits of finitedimensional Lie algebras and their representations, Part III. Collections of unbounded selfadjoint operators satisfying general relations; and Part IV. Representations of operator algebras and noncommutative random sequences.
The theory of *algebras and their representations begins only in the fourth part, where noncommutative probability theory, noncommutative random sequences and noncommutative dynamical systems are introduced, studied and applied to problems in quantum physics.
The result is a fine book, accessible to people familiar with a basic university course in mathematics, including an exposition of the theory of unbounded selfadjoint operators. Some more special results needed for the reading of the book are measure theory on sequence spaces, unitary representations of Lie groups and Lie algebras and some fundamental facts of the theory of *algebras and their representations.
The book appears in excellent typographical conditions, contains a large bibliography and certainly will become and indispensable tool for all interested in mathematical models of quantum systems.