Banach Lattices

This book is mainly concerned with the theory of Banach lattices and with linear operators defined on, or with values in Banach lattices. Moreover we will always consider more general classes of Riesz spaces so long as this does not involve more complicated constructions or proofs. In particular, we will not treat any phenomena which occur only in the non-Banach lattice situation. Riesz spaces, also called vector lattices, K-lineals, are linear lattices which were first considered by F. Riesz, 1. Kantorovic, and H. Freudenthal. Subse quently other important contributions came from the Soviet Union (L.V. Kan torovic, A.J. Judin, A.G. Pinsker, and B.Z. Vulikh), Japan (H. Nakano, T. Oga sawara, and K. Yosida), and the United States (G. Birkhoff, H.F. Bohnenblust, S. Kakutani, and M.l\f. Stone). In the last twenty-five years the theory rapidly increased. Important con tributions came from the Dutch school (W.A.J. Luxemburg, A.C. Zaanen) and the Tiibinger school (lI.lI. Schaefer). In the middle seventies the research on this subject was essentially influenced by the books of H.H. Schaefer (1974) and W.A.J. Luxemburg and A.C. Zaanen (1971). More recently other impor tant books concerning this subject appeared, A.C. Zaanen (1983), H.U. Schwarz (1984), and C.D. Aliprantis and O. Burkinshaw (1985).

Klappentext

This textbook on functional analysis, operator theory and measure theory is intended for advanced students and researchers.

Inhalt
1 Riesz Spaces.- 1.1 Basic Properties of Riesz Spaces and Banach Lattices.- 1.2 Sublattices, Ideals, and Bands.- 1.3 Regular Operators and Order Bounded Functionals.- 1.4 Duality of Riesz Spaces, the Nakano Theory.- 1.5 Extensions of Positive Operators.- 2 Classical Banach Lattices.- 2.1 C(K)-Spaces and M-Spaces.- 2.2 Complex Riesz Spaces.- 2.3 Disjoint Sequences and Approximately Order Bounded Sets.- 2.4 Order Continuity of the Norm, KB-Spaces and the Fatou Property.- 2.5 Weak Compactness.- 2.6 Banach Function Spaces.- 2.7 Lp-Spaces and Related Results.- 2.8 Cone p-Absolutely Summing Operators and p-Subadditive Norms.- 3 Operators on Riesz Spaces and Banach Lattices.- 3.1 Disjointness Preserving Operators and Orthomorphisms on Riesz Spaces.- 3.2 Operators on L-and M-Spaces.- 3.3 Kernel Operators.- 3.4 Order Weakly Compact Operators.- 3.5 Weakly Compact Operators.- 3.6 Approximately Order Bounded Operators.- 3.7 Compact Operators and Dunford-Pettis Operators.- 3.8 Tensor Products of Banach Lattices.- 3.9 Vector Measures and Vectorial Integration.- 4 Spectral Theory of Positive Operators.- 4.1 Spectral Properties of Positive Linear Operators.- 4.2 Irreducible Operators.- 4.3 Measures of Non-Compactness.- 4.4 Local Spectral Theory for Positive Operators.- 4.5 Order Spectrum of Regular Operators.- 4.6 Disjointness Preserving Operators and the Zero-Two Law.- 5 Structures in Banach Lattices.- 5.1 Banach Space Properties of Banach Lattices.- 5.2 Banach Lattices with Subspaces Isomorphic to C(?), C(0,l), and L1(0,1).- 5.3 Grothendieck Spaces.- 5.4 Radon-Nikodym Property in Banach Lattices.- References.