Banach Lattices

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.