H. H. Schaefer, Banach Lattices and Positive Operators (Springer-Verlag, 1974), xi + 376 pp., DM 98.00, $42.70.
A. J. Ellis · Proceedings of the Edinburgh Mathematical Society · 1977
This book gives the first systematic account of positive linear operators acting on Banach lattices.Chapter 1 is devoted to positive square matrices and provides valuable motivation to the remainder of the book.The ideal structure of finite-dimensional vector lattices is used extensively throughout this chapter, in which the Perron-Frobenius theory is presented.The examples of stochastic and doubly stochastic matrices and the applications of the theory to homogeneous Markov chains with finite state space illustrate the material very successfully.Chapter 2 deals with the basic general theory of Banach lattices.The algebraic theory of vector lattices is developed in this chapter, with due emphasis being placed on duality.The special cases of /lZ,-spaces and /4M-spaces are discussed in detail; in particular their duality properties and their representation theorems are fully treated.Complexifications of vector lattices are also introduced here.Chapter 3 begins with properties of closed ideals in Banach lattices and with valuations of vector lattices.This leads to representation theorems for a class of Banach lattices which contains all separable Banach lattices as well as all Banach lattices having an order-continuous norm.The second part of this chapter deals, amongst other things, with mean ergodic theory of semigroups of positive operators and with the representation of compact groups of positive operators on a Banach lattice.Vector lattices of linear operators between Banach lattices are the subject of Chapter 4. Various types of tensor products of Banach lattices are introduced and relationships with the theory of integral maps and absolutely summing maps are shown.Special classes of operators are studied in detail, such as Hilbert-Schmidt operators, nuclear operators, compact operators, and kernel operators between Banach function lattices.The final chapter of the book deals with applications of the earlier material to approximation theory, spectral theory and ergodic theory.The book is written throughout in a clear and attractive style.It has many wellchosen examples, as well as an ample provision of exercises which should prove invaluable to the serious reader.The book is likely to become, and deserves to be, a standard reference work for both the research student and the experienced worker in this field.