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An invitation to operator theory. (English) Zbl 1022.47001
Graduate Studies in Mathematics. 50. Providence, RI: American Mathematical Society (AMS). xiv, 530 p. (2002).
The text offers a thorough introduction to a number of important aspects of operator theory related to order structures. The natural order structures inherent in many Banach spaces provide invaluable insights into the behavior of operators. The book is self-contained, presupposing only the traditional background of introductory graduate courses in analysis and topology. Readers are brought to the forefront of current research in selected topics and thus the book is appropriate for both graduate students and research mathematicians.
The book introduces ordered spaces, Riesz spaces and Banach lattices followed by the various properties of operators on these spaces. These include positive operators, order bounded operators, operators bounded below, continuity properties defined by order, the role of order continuous norms, and the interesting correspondences between compact (and weakly compact) operators and dominated operators. Chapter 3 discusses appropriate abstractions of classical \(L_1\) and \(L_\infty\) spaces for Banach lattices and their consequences. A number of specific types of operators such as multiplicative operators, lattice homomorphisms and Fredholm operators are analyzed. A detailed account of integral operators is included and their relation to the class of abstract integral operators defined in order theoretic terms. Numerous insightful results relate to projections and conditional expectations. A significant portion of the text is concerned with spectral properties of operators with particular emphasis on spectral properties of positive operators and other special classes of operators. A number of consequences for matrices are considered. Generalizing from irreducible matrices, the concept of ideal (or band) irreducible is studied. An operator \(T\) is ideal (band) irreducible if there is no non-trivial closed ideal (band) which is invariant under \(T\). The invariant subspace problem for operators with special order properties (e.g., positive) is considered and the final chapter is devoted to the bounded operators \(T\) satisfying the Daugavet equation, \(\|I+T\|=1+\|T\|\).
The book includes extensive exercises clarifying and extending the text and can be studied in sections since a number of topics are presented independently. The untimely death of the first author is a grievous loss for all who knew him. Yuri Abramovich was a talented and enthusiastic mathematician; his amiable and fervent encouragement was invaluable to many of us.
An accompaying volume to this book presents problems and solutions, see the following review Zbl 1022.47002.

47-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory
46B40 Ordered normed spaces
46B42 Banach lattices
46A40 Ordered topological linear spaces, vector lattices
47B65 Positive linear operators and order-bounded operators
47B60 Linear operators on ordered spaces