POSITIVE OPERATORS

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Dedication To my high school mathematics teachers, ′ Aγγɛλoν Mατζαβ´ινoν και ∆ηµ´ητριoν Kατσ´ωνην Charalambos D. Aliprantis To: Alex, Emily, Joshua, and Andrew Grandchildren bring new joys to your life. Owen Burkinshaw
Contents Foreword ix Historical Foreword xi Preface xiii Acknowledgments xv List of Special Symbols xvii Chapter 1. The Order Structure of Positive Operators 1 §1.1. Basic Properties of Positive Operators 1 §1.2. Extensions of Positive Operators 23 §1.3. Order Projections 31 §1.4. Order Continuous Operators 45 §1.5. Positive Linear Functionals 58 Chapter 2. Components, Homomorphisms, and Orthomorphisms 79 §2.1. The Components of a Positive Operator 80 §2.2. Lattice Homomorphisms 93 §2.3. Orthomorphisms 111 Chapter 3. Topological Considerations 133 §3.1. Topological Vector Spaces 133 §3.2. Weak Topologies on Banach Spaces 156 §3.3. Locally Convex-Solid Riesz Spaces 169 vii
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40 1. The Order Structure of Positi
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Components, Homomorphisms, and Orth
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2.1. The Components of a Positive O
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2.2. Lattice Homomorphisms 93 2.2.
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2.2. Lattice Homomorphisms 95 opera
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2.2. Lattice Homomorphisms 97 Proof
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2.2. Lattice Homomorphisms 99 Theor
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2.2. Lattice Homomorphisms 101 Proo
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2.2. Lattice Homomorphisms 103 comp
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2.2. Lattice Homomorphisms 105 Proo
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2.2. Lattice Homomorphisms 107 comp
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2.2. Lattice Homomorphisms 109 (2)
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2.3. Orthomorphisms 111 (e) Present
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2.3. Orthomorphisms 113 T (y) ∈ B
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2.3. Orthomorphisms 115 Proof. Let
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2.3. Orthomorphisms 117 The disjoin
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2.3. Orthomorphisms 119 defines an
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2.3. Orthomorphisms 121 follows fro
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2.3. Orthomorphisms 123 case we hav
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2.3. Orthomorphisms 125 Proof. We c
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2.3. Orthomorphisms 127 be identifi
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2.3. Orthomorphisms 129 is R X . By
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2.3. Orthomorphisms 131 6. Let (P)
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Chapter 3 Topological Consideration
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3.1. Topological Vector Spaces 135
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3.2. Weak Topologies on Banach Spac
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3.3. Locally Convex-Solid Riesz Spa
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Banach Lattices Chapter 4 It is wel
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4.1. Banach Lattices with Order Con
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4.2. Weak Compactness in Banach Lat
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4.3. Embedding Banach Spaces 221 (a
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4.3. Embedding Banach Spaces 223 th
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4.3. Embedding Banach Spaces 225 (3
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4.3. Embedding Banach Spaces 227 th
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4.3. Embedding Banach Spaces 229 (b
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4.3. Embedding Banach Spaces 231 Th
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4.3. Embedding Banach Spaces 233 It
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4.3. Embedding Banach Spaces 235 (1
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4.3. Embedding Banach Spaces 237 ˙
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4.3. Embedding Banach Spaces 239 C
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4.3. Embedding Banach Spaces 241 Pr
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4.3. Embedding Banach Spaces 243 ho
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4.3. Embedding Banach Spaces 245 In
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4.3. Embedding Banach Spaces 247 Le
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4.3. Embedding Banach Spaces 249 ST
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4.3. Embedding Banach Spaces 251 co
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4.3. Embedding Banach Spaces 253 Th
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4.4. Banach Lattices of Operators 2
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Compactness Properties of Positive
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5.1. Compact Operators 275 An opera
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5.1. Compact Operators 277 Proof. L
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5.1. Compact Operators 279 holds fo
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5.1. Compact Operators 281 simplici
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5.1. Compact Operators 283 To this
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5.1. Compact Operators 285 and S2(
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5.1. Compact Operators 287 “roots
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5.1. Compact Operators 289 10. (Loz
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5.2. Weakly Compact Operators 291 (
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5.2. Weakly Compact Operators 293 d
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5.2. Weakly Compact Operators 295 P
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5.2. Weakly Compact Operators 297 N
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5.2. Weakly Compact Operators 299 T
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5.2. Weakly Compact Operators 301 s
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5.2. Weakly Compact Operators 303 F
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5.2. Weakly Compact Operators 305 a
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5.2. Weakly Compact Operators 307 J
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5.2. Weakly Compact Operators 309 b
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5.2. Weakly Compact Operators 311 U
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5.2. Weakly Compact Operators 313 (
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5.2. Weakly Compact Operators 315 w
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5.3. L-and M-weakly Compact Operato
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5.4. Dunford-Pettis Operators 341 T
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5.4. Dunford-Pettis Operators 343 t
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5.4. Dunford-Pettis Operators 345 w
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5.4. Dunford-Pettis Operators 347 P
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5.4. Dunford-Pettis Operators 349 N
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5.4. Dunford-Pettis Operators 351 y
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5.4. Dunford-Pettis Operators 353 N
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5.4. Dunford-Pettis Operators 355 c
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5.4. Dunford-Pettis Operators 357 1
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360 Bibliography 15. C. D. Aliprant
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362 Bibliography 61. P. Enflo, A co
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364 Bibliography 104. M. G. Krein a
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366 Bibliography 144. L. C. Moore,
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368 Bibliography 188. A. I. Veksler
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Index AL-space, 194 AM-space, 193,
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Index 373 supporting, 138 gauge, 13
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Index 375 projection band, 35 proje
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POSITIVE OPERATORS

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