Banach Lattices

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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.lf. 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).


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


1 Riesz Spaces.- 1.1 Basic Properties of Riesz Spaces and Banach Lattices.- Elementary Properties of Ordered Spaces.- Elementary Properties of Riesz Spaces.- Normed Riesz Spaces, Definition.- Order-Completeness Properties of Riesz Spaces.- Order Convergence.- 1.2 Sublattices, Ideals, and Bands.- Definition and Elementary Properties.- Bands and Band Projections.- Order Units, M-Norms, and M-Spaces.- Freudenthal's Spectral Theorem and Quasi Units.- 1.3 Regular Operators and Order Bounded Functionals.- Positive and Regular Operators.- Regular Operators on Banach Lattices, the r-Norm.- Order Continuous Operators.- Lattice Homomorphisms.- 1.4 Duality of Riesz Spaces, the Nakano Theory.- Elementary Duality Results.- Embedding of E into E" as a Sublattice.- L-Spaces.- Carrier of Positive Functionals.- Embedding of E into E" as an Ideal, the Nakano Theory.- Characterization of Lattice Homomorphisms by Duality.- 1.5 Extensions of Positive Operators.- Sublinear Operators and the Hahn-Banach Theorem.- Extensions of Positive Operators.- Extensions of Lattice Homomorphisms.- 2 Classical Banach Lattices.- 2.1 C(K)-Spaces and M-Spaces.- The Stone-Weierstraß Theorem.- Kakutani's Representation Theorem for M-Spaces.- Characterization of Dedekind Complete C(K)-Spaces.- Hyper-Stonian Spaces, Dixmier's Theorem.- Characterization of Closed Ideals and Bands of C(K).- Characterization of M-Spaces.- Embeddings of Banach Spaces into ?? or C(?).- Extension of Continuous Functions.- A Model for Uniformly Complete Riesz Spaces.- 2.2 Complex Riesz Spaces.- Complexification of Uniformly Complete Riesz Spaces.- Complexification of Banach Lattices.- Complex Regular Operators.- 2.3 Disjoint Sequences and Approximately Order Bounded Sets.- Constructions of Disjoint Sequences.- The Disjoint Sequence Theorem.- Rosenthal's Lemma.- Sublattice Embeddings of c0, ?1 and ??.- 2.4 Order Continuity of the Norm, KB-Spaces and the Fatou Property.- Characterizations of Order Continuous Norms.- Order Topology.- Amimeya's Theorem.- KB-Spaces and Reflexive Banach Lattices.- The Fatou Property.- 2.5 Weak Compactness.- Properties of Weakly Sequentially Precompact Sets.- The Dunford-Pettis Theorem.- Weak Compactness in the Space of Radon Measures.- Weakly*-Sequentially Precompact Sets.- Weakly Sequentially Precompact Sets.- Grothendieck's ??-Theorem.- Phillip's Lemma.- Convergence Theorems for Sequences of Measures.- 2.6 Banach Function Spaces.- Definition and Preliminary Results.- The Riesz-Fischer Property.- Associate Spaces and Norms.- Luxemburg Norms and Young Functions.- Orlicz Spaces.- 2.7 Lp-Spaces and Related Results.- Kakutani's Representation Theorem for Lp-Spaces.- Classifications of Separable Lp-Spaces.- Hilbert Lattices.- Khinchine's Inequalities.- Representation of Banach Lattices as Ideals in L1(?).- Bohnenblust's Characterization of p-Additive Norms.- Lp-Spaces and Contractive Projections, Ando's Theorem.- 2.8 Cone p-Absolutely Summing Operators and p-Subadditive Norms.- p-Super additive and p-Subadditive Norms.- Cone p-Absolutely Summing and p-Majorizing Operators.- Factorization of p-Absolutely Summing Operators.- Characterization of p-Absolutely Summing Operators.- 3 Operators on Riesz Spaces and Banach Lattices.- 3.1 Disjointness Preserving Operators and Orthomorphisms on Riesz Spaces.- Definitions and Elementary Results.- The Modulus of a Regular Disjointness Preserving Operator.- Regularity of Disjointness Preserving Operators.- Properties of Orthomorphisms.- f-Algebras and Orthomorphisms.- Characterization of the Center.- Representation of Majorized Operators.- Projection onto the Center.- Approximation of Components of Operators.- 3.2 Operators on L-and M-Spaces.- Characterization of L- and M- Spaces.- Injective Banach Lattices.- Lattice Homomorphisms on Spaces of Type C(K).- Norm Identities for Operators on L- and M- Spaces.- 3.3 Kernel Operators.- Elementary Properties of Kernel Operators.- Operators Majorized by Kernel Operators.- The Band of Kernel Operators.- A Characterization of Kernel Operators.- Dunford's Theorem.- 3.4 Order Weakly Compact Operators.- Characterization of Order Weakly Compact Operators.- Factorization of Order Weakly Compact Operators.- Operators Preserving No Subspaces Isomorphic to c0.- Order Weakly Compact Dual Operators.- Weakly Sequentially Precompact Operators.- 3.5 Weakly Compact Operators.- Interpolation Space for an Operator.- Factorization of Weakly Compact Operators.- Permanence Properties of Weakly Compact Operators.- The Space of all Weakly Compact Operators.- 3.6 Approximately Order Bounded Operators.- L-Weakly Compact Subsets.- Semicompact and L-Weakly Compact Operators.- M-Weakly Compact Operators.- L-Weakly Compact Regular Operators.- 3.7 Compact Operators and Dunford-Pettis Operators.- AM-Compact Operators.- Compactness of AM-Compact Operators.- Dunford-Pettis Spaces and Operators.- The Reciprocal Dunford-Pettis Property.- Permanence Properties of Compact Operators.- Permanence Properties of Dunford-Pettis Operators.- The Space of Dunford-Pettis Operators.- 3.8 Tensor Products of Banach Lattices.- Approximation Property of Lp- and C(K)-Spaces.- Regularly Ordered Tensor Products.- Tensor Products of Banach Lattices.- Special Tensor Norms.- 3.9 Vector Measures and Vectorial Integration.- Countably and Strongly Additive Vector Measures.- Characterization of Strongly Additive Vector Measures.- Absolute Continuity.- ?-Measurable X-Valued Functions.- Bochner Integrable Functions.- 4 Spectral Theory of Positive Operators.- 4.1 Spectral Properties of Positive Linear Operators.- Positive Resolvents.- Power Series with Positive Coefficients.- Krein-Rutman Theorems.- Embedding a Banach Lattice into an Ultra-Product.- Spectrum of Lattice Homomorphisms.- Operators with Cyclic Spectrum.- Lower Bounds for Positive Operators.- 4.2 Irreducible Operators.- Topological Nilpotency of Irreducible Operators.- Compact Irreducible Operators.- Band Irreducible Operators.- Multiplicity of Eigenvalues of Irreducible Operators.- 4.3 Measures of Non-Compactness.- A Formula for the Measure of Non-Compactness.- Interval Preserving Operators and Lattice Homomorphisms.- Fredholm Operators and the Measure of Non-Compactness.- Essential Spectral Radius for AM-Compact Operators.- 4.4 Local Spectral Theory for Positive Operators.- Local Spectral Radius and Resolvent.- Positive Solutions of (?I - T)z = x.- Chain of Invariant Ideals.- Minimal Value of an Operator.- 4.5 Order Spectrum of Regular Operators.- Characterization of the Order Spectrum.- Operators Satisfying ?o(T) = ?(T).- An Operator Satisfying ?o(T) ? ?(T).- 4.6 Disjointness Preserving Operators and the Zero-Two Law.- Power Bounded Operators.- Spectrum and Power Bounded Operators.- The Zero-Two Law.- Spectrum of Disjointness Preserving Operators.- 5 Structures in Banach Lattices.- 5.1 Banach Space Properties of Banach Lattices.- Subspace Embeddings of cO.- The James Space J.- Banach Lattices with Property (u).- Complemented Subspaces of Banach Lattices.- 5.2 Banach Lattices with Subspaces Isomorphic to C(?), C(0,l), and L1(0,1).- Subsets Homeomorphic to the Cantor Set.- Operators not Preserving Subspaces Isomorhic to ?1.- Sublattices Isomorphic to L1 (0,1).- 5.3 Grothendieck Spaces.- Property (V) and (V*).- Property (V0).- Characterization of Grothendieck Spaces.- Operators Preserving Subspaces Isomorphic to C(?).- 5.4 Radon-Nikodym Property in Banach Lattices.- Representable Operators and the Radon-Nikodym Property.- Spaces without the Radon-Nikodym Property.- Spaces Possessing the Radon-Nikodym Property.- Dual Banach Lattices with the Radon-Nikodym Property.- Order Dentable Banach Lattices.- Order Dentable Spaces and the Radon-Nikodym Property.- Characterization of Separable Dual Banach Lattices.- References.

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Banach Lattices
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Springer Berlin Heidelberg
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