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Banach lattices. (English) Zbl 0743.46015

Universitext. Berlin etc.: Springer-Verlag. xv, 395 p. (1991).
Since the publication of the book “Banach lattices and positive operators” of H. .H. Schaefer in 1974 ( Zbl 0296.47023) the whole theory has been rapidly developed up to now. From time to time the most relevant new results were collected together in monographs, see e.g. A. C. Zaanen: Riesz spaces. II (1983; Zbl 0519.46001); H. U. Schwarz: Banach lattices and operators (1984; Zbl 0585.47025); C. D. Aliprantis and O. Burkinshaw: Positive operators (1985; Zbl 0608.47039).
Since the publication of this last book a lot of important contributions were made to the theory, e.g. de Pagter’s result on irreducible compact operators on Banach lattices, the results of Ghoussub and Johnson on the factorization of operators through Banach lattices not containing \(C[0,1]\), Voigt’s result on uniform continuity of the projection of the space of regular operators onto the center, a lot of results on weakly compact and related operators on Banach lattices, Räbiger’s contributions to the structure of Grothendieck-spaces, and so on. So the publication of a new monograph is completely justified.
The book under review satisfies all the requirements on the integration of these and a lot of more results in an ideal manner. The list of references containing around fifty important items published since 1985, including some preprints.
The author is very well-known because of his contributions to the characterization of Banach lattices by means of special sequences. The methods he developed in the early seventieths are used in this book to simplify proofs of known deep theorems as well as to improve other results considerably.
The representation of the material is always very elegant, so it is a great pleasure to read the book on the whole as well as to use it as a source of information about the new developments in the theory.
The book is divided into five chapters. The first one contains the elementary theory of vector lattices including the lattice-valued version of Hahn-Banach’s theorem. Its proof, based on König’s technique of sublinearity, as well as its applications are a first highlight in the text.
The second chapter is mainly devoted to the theory of disjoint sequences. As such it is basic to the rest of the book. First applications are devoted to weak compactness of sets in Banach lattices including classical results on \(L^ p\)-spaces.
The third chapter is concerned essentially with regular operators on Banach lattices and factorization theorems (including various types of weakly compact and related operators). The fourth chapter is devoted to global and local spectral theory of a single operator, including de Pager’s theorem mentioned above as well as measures of noncompactness and results on the essential spectrum.
Finally in the last chapter the deep results on the structure of Banach lattices published since 1985 are treated (Grothendieck spaces and many other things).
At the end of each section the interested reader will find several exercises.
There are a few minor points to be criticized: first of all there is no author index, secondly one misses also an index of symbols. Furthermore it is not quite clear which criteria are used in honouring authors by naming theorems after them (compare Voigt’s important theorem 3.1.22).
Finally no applications to problems of hard concrete analysis are mentioned. But as pointed out above these points of critique are only of minor importance. I enjoy to recommend this monograph to experts as well as to experienced students.

MSC:

46B42 Banach lattices
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
47B65 Positive linear operators and order-bounded operators
47B07 Linear operators defined by compactness properties
46A40 Ordered topological linear spaces, vector lattices
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