Banach lattices

Peter Meyer-Nieberg9783540542018, 3540542019, 0387542019, 9780387542010

This book is concerned primarily with the theory of Banach lattices and with linear operators defined on, or with values in, Banach lattices. More general classes of Riesz spaces are considered so long as this does not lead to more complicated constructions or proofs. The intentions for writing this book were twofold. First, there appeared in the literature many results completing the theory extensively. On the other hand, new techniques systematically applied here for the first time lead to surprisingly simple and short proofs of many results originally known as deep. These new methods are purely elementary: they directly yield the Banach lattice versions of theorems which then include the classical theorems in a trivial manner. In particular the book covers: Riesz spaces, normed Riesz spaces, C(K)-and Mspaces, Banach function spaces, Lpspaces, tensor products of Banach lattices, Grothendieck spaces; positive and regular operators, extensions of positive operators, disjointness-preserving operators, operators on L- and M-spaces, kernel operators, weakly compact operators and generalizations, Dunford-Pettis operators and spaces, irreducible operators; order continuity of norms, p-subadditive norms; spectral theory, order spectrum; embeddings of C; the Radon-Nikodym property; measures of non-compactness. This textbook on functional analysis, operator theory and measure theory is intended for advanced students and researchers.

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Table of contents :
Cover ……Page 1
Series ……Page 2
Title page ……Page 3
Date-line ……Page 4
Preface ……Page 5
Contents ……Page 11
Elementary Properties of Ordered Spaces ……Page 17
Elementary Properties of Riesz Spaces ……Page 18
Normed Riesz Spaces, Definition ……Page 22
Order-Completeness Properties of Riesz Spaces ……Page 23
Order Convergence ……Page 25
Definition and Elementary Properties ……Page 28
Bands and Band Projections ……Page 30
Order Units, M-Norms, and M-Spaces ……Page 34
Freudenthal’s Spectral Theorem and Quasi Units ……Page 36
Positive and Regular Operators ……Page 40
Regular Operators on Banach Lattices, the r-Norm ……Page 43
Order Continuous Operators ……Page 44
Lattice Homomorphisms ……Page 46
Elementary Duality Results ……Page 48
Embedding of $E$ into $E”$ as a Sublattice ……Page 50
L-Spaces ……Page 51
Carrier of Positive Functionals ……Page 52
Embedding of $E$ into $E”$ as an Ideal, the Nakano Theory ……Page 54
Characterization of Lattice Homomorphisms by Duality ……Page 57
Sublinear Operators and the Hahn-Banach Theorem ……Page 59
Extensions of Positive Operators ……Page 62
Extensions of Lattice Homomorphisms ……Page 65
The Stone-Weierstrass Theorem ……Page 67
Kakutani’s Representation Theorem for M-Spaces ……Page 69
Characterization of Dedekind Complete $C(K)$-Spaces ……Page 70
Hyper-Stonian Spaces, Dixmier’s Theorem ……Page 72
Characterization of Closed Ideals and Bands of $C(K)$ ……Page 73
Characterization of M-Spaces ……Page 75
Extension of Continuous Functions ……Page 78
A Model for Uniformly Complete Riesz Spaces ……Page 82
Complexification of Uniformly Complete Riesz Spaces ……Page 83
Complexification of Banach Lattices ……Page 84
Complex Regular Operators ……Page 86
Constructions of Disjoint Sequences ……Page 87
The Disjoint Sequence Theorem ……Page 91
Rosenthal’s Lemma ……Page 94
Sublattice Embeddings of $c_0$, $mathcal{l}^1$, and $mathcal{l}^infty$ ……Page 98
Characterizations of Order Continuous Norms ……Page 102
Order Topology ……Page 105
Amimeya’s Theorem ……Page 107
KB-Spaces and Reflexive Banach Lattices ……Page 108
The Fatou Property ……Page 112
Properties of Weakly Sequentially Precompact Sets ……Page 115
The Dunford-Pettis Theorem ……Page 117
Weak Compactness in the Space of Radon Measures ……Page 118
Weakly$^ast$-Sequentially Precompact Sets ……Page 121
Weakly Sequentially Precompact Sets ……Page 122
Grothendieck’s $mathcal{l}^infty$-Theorem ……Page 127
Convergence Theorems for Sequences of Measures ……Page 128
Definition and Preliminary Results ……Page 130
The Riesz-Fischer Property ……Page 132
Associate Spaces and Norms ……Page 133
Luxemburg Norms and Young Functions ……Page 136
Orlicz Spaces ……Page 137
Kakutani’s Representation Theorem for $L^p$-Spaces ……Page 140
Classifications of Separable $L^p$-Spaces ……Page 141
Khinchine’s Inequalities ……Page 144
Representation of Banach Lattices as Ideals in $L^1(mu)$ ……Page 146
Bohnenblust’s Characterization of p-Additive Norms ……Page 149
$L^p$-Spaces and Contractive Projections, Ando’s Theorem ……Page 150
p-Superadditive and p-Subadditive Norms ……Page 154
Cone p-Absolutely Summing and p-Majorizing Operators ……Page 156
Factorization of p-Absolutely Summing Operators ……Page 159
Characterization of p-Absolutely Summing Operators ……Page 160
Definitions and Elementary Results ……Page 165
The Modulus of a Regular Disjointness Preserving Operator ……Page 166
Regularity of Disjointness Preserving Operators ……Page 168
Properties of Orthomorphisms ……Page 170
$f$-Algebras and Orthomorphisms ……Page 171
Characterization of the Center ……Page 173
Representation of Majorized Operators ……Page 177
Projection onto the Center ……Page 180
Approximation of Components of Operators ……Page 181
Characterization of L- and M- Spaces ……Page 184
Injective Banach Lattices ……Page 186
Lattice Homomorphisms on Spaces of Type C(K) ……Page 188
Norm Identities for Operators on L- and M- Spaces ……Page 190
Elementary Properties of Kernel Operators ……Page 192
Operators Majorized by Kernel Operators ……Page 194
The Band of Kernel Operators ……Page 197
A Characterization of Kernel Operators ……Page 202
Dunford’s Theorem ……Page 206
3.4 Order Weakly Compact Operators ……Page 207
Characterization of Order Weakly Compact Operators ……Page 208
Factorization of Order Weakly Compact Operators ……Page 209
Operators Preserving No Subspaces Isomorphic to $c_0$ ……Page 212
Order Weakly Compact Dual Operators ……Page 213
Weakly Sequentially Precompact Operators ……Page 216
Interpolation Space for an Operator ……Page 219
Factorization of Weakly Compact Operators ……Page 222
Permanence Properties of Weakly Compact Operators ……Page 224
The Space of all Weakly Compact Operators ……Page 225
3.6 Approximately Order Bounded Operators ……Page 227
L-Weakly Compact Subsets ……Page 228
M-Weakly Compact Operators ……Page 229
L-Weakly Compact Regular Operators ……Page 231
AM-Compact Operators ……Page 234
Dunford-Pettis Spaces and Operators ……Page 235
The Reciprocal Dunford-Pettis Property ……Page 237
Permanence Properties of Compact Operators ……Page 238
Permanence Properties of Dunford-Pettis Operators ……Page 240
The Space of Dunford-Pettis Operators ……Page 242
3.8 Tensor Products of Banach Lattices ……Page 245
Approximation Property of $L^p$- and $C(K)$-Spaces ……Page 246
Regularly Ordered Tensor Products ……Page 247
Tensor Products of Banach Lattices ……Page 250
Special Tensor Norms ……Page 251
Countably and Strongly Additive Vector Measures ……Page 254
Characterization of Strongly Additive Vector Measures ……Page 255
Absolute Continuity ……Page 257
$lambda$-Measurable $X$-Valued Functions ……Page 258
Bochner Integrable Functions ……Page 259
4.1 Spectral Properties of Positive Linear Operators ……Page 263
Positive Resolvents ……Page 264
Power Series with Positive Coefficients ……Page 265
Krein-Rutman Theorems ……Page 266
Embedding a Banach Lattice into an Ultra-Product ……Page 268
Spectrum of Lattice Homomorphisms ……Page 270
Operators with Cyclic Spectrum ……Page 272
Lower Bounds for Positive Operators ……Page 275
4.2 Irreducible Operators ……Page 277
Topological Nilpotency of Irreducible Operators ……Page 278
Compact Irreducible Operators ……Page 280
Band Irreducible Operators ……Page 283
Multiplicity of Eigenvalues of Irreducible Operators ……Page 288
4.3 Measures of Non-Compactness ……Page 290
A Formula for the Measure of Non-Compactness ……Page 294
Interval Preserving Operators and Lattice Homomorphisms ……Page 296
Fredholm Operators and the Measure of Non-Compactness ……Page 299
Essential Spectral Radius for AM-Compact Operators ……Page 301
4.4 Local Spectral Theory for Positive Operators ……Page 303
Local Spectral Radius and Resolvent ……Page 304
Positive Solutions of $(lambda I – T)z = x$ ……Page 306
Chain of Invariant Ideals ……Page 309
Minimal Value of an Operator ……Page 311
Characterization of the Order Spectrum ……Page 317
Operators Satisfying $sigma_o(T) = sigma(T)$ ……Page 318
An Operator Satisfying $sigma_o(T) neq sigma(T)$ ……Page 321
4.6 Disjointness Preserving Operators and the Zero-Two Law ……Page 325
Power Bounded Operators ……Page 327
Spectrum and Power Bounded Operators ……Page 328
The Zero-Two Law ……Page 330
Spectrum of Disjointness Preserving Operators ……Page 332
5.1 Banach Space Properties of Banach Lattices ……Page 337
Subspace Embeddings of $c_0$ ……Page 339
The James Space $J$ ……Page 341
Banach Lattices with Property (u) ……Page 343
Complemented Subspaces of Banach Lattices ……Page 345
Subsets Homeomorphic to the Cantor Set ……Page 347
Operators not Preserving Subspaces Isomorhic to $mathcal{l}^1$ ……Page 359
Sublattices Isomorphic to $L^1(0,1)$ ……Page 361
5.3 Grothendieck Spaces ……Page 364
Property (V) and (V$^ast$) ……Page 365
Property (V$_0$) ……Page 368
Characterization of Grothendieck Spaces ……Page 369
Operators Preserving Subspaces Isomorphic to $C(Delta)$ ……Page 375
Representable Operators and the Radon-Nikodym Property ……Page 376
Spaces without the Radon-Nikodym Property ……Page 378
Spaces Possessing the Radon-Nikodym Property ……Page 379
Dual Banach Lattices with the Radon-Nikodym Property ……Page 383
Order Dentable Banach Lattices ……Page 384
Characterization of Separable Dual Banach Lattices ……Page 388
References ……Page 395
Index ……Page 409