Banach Lattices (Universitext)

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. 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 This text uses Banach lattices and Riesz spaces as tools to study the classical ideas and theorems of functional analysis. It is suitable for students who have completed a standard course in functional analysis and for mathematicians interested in the systematic application of such techniques.