Vector Lattices and Intergal Operators

The theory of vector lattices, stemming from the mid-thirties, is now at the stage where its main achievements are being summarized. The sweeping changes of the last two decades have changed its image completely. The range of its application was expanded and enriched so as to embrace diverse branches of the theory of functions, geometry of Banach spaces, operator theory, convex analysis, etc. Furthermore, the theory of vector lattices was impregnated with principally new tools and techniques from other sections of mathematics. These circumstances gave rise to a series of mono graphs treating separate aspects of the theory and oriented to specialists. At the same time, the necessity of a book intended for a wider readership, reflecting the modern diretions of research became clear. The present book is meant to be an attempt at implementing this task. Although oriented to readers making their first acquaintance with vector-lattice theory, it is composed so that the main topics dealt with in the book reach the current level of research in the field, which is of interest and import for specialists. The monograph was conceived so as to be divisible into two parts that can be read independently of one another. The first part is mainly Chapter 1, devoted to the so-called Boolean-valued analysis of vector lattices. The term designates the applica tion of the theory of Boolean-valued models by D. Scott, R. Solovay and P.

Klappentext

This volume is devoted to modern accomplishments in the field of vector lattices and integral operators which were achieved in Russia during the last two decades. Nonstandard methods are elaborated for the analysis of vector lattices and positive operators. Much attention is paid to studying stability under multiplication by an arbitrary bounded operator for various classes of operators which are defined in terms of order. Also, several approaches to the solution of the J. von Neumann problem on the conditions for integrability of a linear operator are treated, and full information is given on the solution of some problems posed by P. Halmos and V. Sunder. Audience: This book is intended for mathematicians, students and postgraduates interested in functional analysis, operator theory, geometry of Banach spaces and vector lattices.

Inhalt

1. Nonstandard Theory of Vector Lattices.- 1. Vector Lattices.- 2. Boolean-Valued Models.- 3. Real Numbers in BooleanValued Models.- 4. Boolean-Valued Analysis of Vector Lattices.- 5. Fragments of Positive Operators.- 6. LatticeNormed Spaces.- Comments.- References.- 2. Operator Classes Determined by Order Conditions.- 1. Ideal Spaces.- 2. The Space of Regular Operators.- 3. Spaces of Vector-Functions.- 4. Dominated Operators.- Comments.- References.- 3. Stably Dominated and Stably Regular Operators.- 1. pAbsolutely Summing Operators.- 2. pAbsolutely Summing Operators in Hilbert Space.- 3. Nuclear Operators.- 4. Stably Dominated Operators.- 5. Coincidence of Some Classes of Operators in the Scale of the Banach Spaces LP.- 6. NikishinMaurey Factorization Theorems.- 7. Stably Regular Operators.- 8. Certain Operator Lattices.- 9. Operator Spaces and Local Unconditional Structure.- S. Supplement to Chapter 3.- Comments.- References.- 4. Integral Operators.- 1. Basic Properties of Integral Operators.- 2. Integral Representation of Linear Operators.- 3. Applications of the Criterion for Integral Representability.- 4. Linear Operators and Vector Measures.- 5. Integral Representation of Nonlinear Operators.- 6. Algebraic Properties of Integral Operators.- 7. Universal Integral Operators and Operators with Integral Commutators.- Comments.- References.- S. Supplement to Chapter 4. Integral Operators of Convolution (V.D. Stepanov).- References.- 5. Disjointness Preserving Operators.- 1. Prerequisites.- 2. Order Approximating Sets.- 3. Order Bounded Operators.- 4. The Shadow of an Operator.- 5. Orthomorphisms.- 6. Shift Operators.- 7. Weighted Shift Operator.- 8. Representation of Disjointness Preserving Operators.- 9. Interpretation for the Properties of Operators.-Comments.- References.