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**Positive operators.**
*(English)*
Zbl 0608.47039

Pure and Applied Mathematics, 119. Orlando etc.: Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers). XVI, 367 p. $ 59.00; £51.50 (1985).

The book is devoted to positive and, more generally, order bounded operators on Riesz spaces and Banach lattices. Also, related fragments of the theory of continuous operators on Banach spaces (without order structure) are presented. The authors concentrate on results obtained in the 1970s and 1980s. Accordingly, a great deal of the contents of the book, including many results due to the authors, makes here its first appearance in book form. Some of the more classical results have also been incorporated, occasionally without proof, thereby increasing the usefulness of the book for the less experienced reader.

The material is divided into five chapters: 1. The order structure of positive operators; 2. Components, homomorphisms, and orthomorphisms; 3. Topological considerations; 4. Banach lattices; 5. Compactness properties of positive operators. The contents of the chapters are described by the authors in the Preface as follows: ”Chapter 1 deals mainly with the elementary properties of positive operators. This chapter covers extension properties of positive operators, order projections, order continuous operators, and positive linear functionals. Chapter 2 studies three specific classes of operators: the components of a positive operator, the lattice homomorphisms, and the orthomorphisms. Chapter 3 considers topological aspects of vector spaces. It covers topological vector spaces, weak topologies on Banach and Riesz spaces, and locally convex-solid Riesz spaces. The fourth chapter is devoted to Banach lattices. Particular emphasis is given to Banach lattices with order continuous norms. Also, weak compactness in Banach lattices, embeddings of Banach lattices, and Banach lattices of operators are studied in this chapter. The fifth and final chapter of the book deals with compactness properties of positive operators. This is the most important (and most elegant) chapter of the book. It makes a thorough study of compact, weakly compact, and Dunford-Pettis operators on Banach lattices.” In addition, the book contains almost 300 exercises, ranging from nearly trivial to quite challenging ones.

The style of presentation is lively and conveys some of the authors’ enthusiasm for the subject. Misprints appear only occasionally, not counting the titles of German-language reference items, where the initial letters of nouns are usually small. In sum, the book under review is a welcome addition to the fast growing literature of Riesz spaces and Banach lattices and operators on them. The almost simultaneous appearance of three other related books: S. Kaplan, The bidual of C(X), Amsterdam (1985; Zbl 0583.46004), A. G. Kusraev, Vector duality and its applications (in Russian) (1985) and H.-U. Schwarz, Banach lattices and operators (1984; Zbl 0585.47025) testifies to the current popularity of the subject.

Five earlier books also belonging to this field: G. P. Akilov and S. S. Kutateladze, Ordered vector spaces (in Russian) (1978; Zbl 0395.46010), R. Cristescu, Ordered vector spaces and linear operators (1976; Zbl 0322.46010), E. de Jonge and A. C. M. van Rooij, Introduction to Riesz spaces (1977; Zbl 0421.46001), B. Z. Vulikh, Introduction to the theory of cones in normed spaces (in Russian) (1977), and B. Z. Vulikh, Special questions in the geometry of cones in normed spaces (in Russian) (1978) do not appear in the bibliography of the book under review either.

The material is divided into five chapters: 1. The order structure of positive operators; 2. Components, homomorphisms, and orthomorphisms; 3. Topological considerations; 4. Banach lattices; 5. Compactness properties of positive operators. The contents of the chapters are described by the authors in the Preface as follows: ”Chapter 1 deals mainly with the elementary properties of positive operators. This chapter covers extension properties of positive operators, order projections, order continuous operators, and positive linear functionals. Chapter 2 studies three specific classes of operators: the components of a positive operator, the lattice homomorphisms, and the orthomorphisms. Chapter 3 considers topological aspects of vector spaces. It covers topological vector spaces, weak topologies on Banach and Riesz spaces, and locally convex-solid Riesz spaces. The fourth chapter is devoted to Banach lattices. Particular emphasis is given to Banach lattices with order continuous norms. Also, weak compactness in Banach lattices, embeddings of Banach lattices, and Banach lattices of operators are studied in this chapter. The fifth and final chapter of the book deals with compactness properties of positive operators. This is the most important (and most elegant) chapter of the book. It makes a thorough study of compact, weakly compact, and Dunford-Pettis operators on Banach lattices.” In addition, the book contains almost 300 exercises, ranging from nearly trivial to quite challenging ones.

The style of presentation is lively and conveys some of the authors’ enthusiasm for the subject. Misprints appear only occasionally, not counting the titles of German-language reference items, where the initial letters of nouns are usually small. In sum, the book under review is a welcome addition to the fast growing literature of Riesz spaces and Banach lattices and operators on them. The almost simultaneous appearance of three other related books: S. Kaplan, The bidual of C(X), Amsterdam (1985; Zbl 0583.46004), A. G. Kusraev, Vector duality and its applications (in Russian) (1985) and H.-U. Schwarz, Banach lattices and operators (1984; Zbl 0585.47025) testifies to the current popularity of the subject.

Five earlier books also belonging to this field: G. P. Akilov and S. S. Kutateladze, Ordered vector spaces (in Russian) (1978; Zbl 0395.46010), R. Cristescu, Ordered vector spaces and linear operators (1976; Zbl 0322.46010), E. de Jonge and A. C. M. van Rooij, Introduction to Riesz spaces (1977; Zbl 0421.46001), B. Z. Vulikh, Introduction to the theory of cones in normed spaces (in Russian) (1977), and B. Z. Vulikh, Special questions in the geometry of cones in normed spaces (in Russian) (1978) do not appear in the bibliography of the book under review either.

Reviewer: Z.Lipecki

### MSC:

47B60 | Linear operators on ordered spaces |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

46A40 | Ordered topological linear spaces, vector lattices |

46B42 | Banach lattices |