Narrow Operators on Function Spaces and Vector Lattices (eBook)

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Mikhail Popov, Chernivtsi National University, Ukraine; Miami University, Oxford, USA; Beata Randrianantoanina, Miami University, Oxford, USA.

Preface 7
1 Introduction and preliminaries 15
1.1 Background information 15
1.2 Terminology and notation 15
1.3 Narrow operators on function spaces 16
1.4 Homogeneous measure spaces and Maharam’s theorem 23
1.5 Necessary information on vector lattices 26
1.6 Kalton’s and Rosenthal’s representation theorems for operators on L1 35
2 Each “small” operator is narrow 38
2.1 AM-compact and Dunford-Pettis operators are narrow 38
2.2 “Large” subspaces are exactly strictly rich 40
2.3 Operators with “small” ranges are narrow 42
2.4 Narrow operators are compact on a suitable subspace 46
3 Applications to nonlocally convex spaces 50
3.1 Nonexistence of nonzero narrow operators 51
3.2 The separable quotient space problem 52
3.3 Isomorphic classification of strongly nonconvex Köthe F-spaces 52
4 Noncompact narrow operators 55
4.1 Conditional expectation operators 56
4.2 A narrow projection from E onto a subspace isometric to E 57
4.3 A characterization of narrow conditional expectations 65
5 Ideal properties, conjugates, spectrum and numerical radii 71
5.1 Ideal properties of narrow operators and stability of rich subspaces 71
5.2 Conjugates of narrow operators need not be narrow 74
5.3 Spectrum of a narrow operator 74
5.4 Numerical radii of narrow operators on Lp (µ)-spaces 76
6 Daugavet-type properties of Lebesgue and Lorentz spaces 85
6.1 A generalization of the DP for L1 to “small” into isomorphisms 86
6.2 Pseudo-Daugavet property for narrow operators on Lp, p . 2 91
6.3 A pseudo-Daugavet property for narrow projections in Lorentz spaces 103
6.4 Near isometric classification of Lp (µ)-spaces for 1 = p< 8, p . 2
7 Strict singularity versus narrowness 123
7.1 Bourgain-Rosenthal’s theorem on l1 -strictly singular operators 125
7.2 Rosenthal’s characterization of narrow operators on L1 146
7.3 Johnson-Maurey-Schechtman-Tzafriri’s theorem 164
7.4 An application to almost isometric copies of L1 185
7.5 An application to complemented subspaces of Lp 187
7.6 The Daugavet property for rich subspaces of L1 188
8 Weak embeddings of L1 193
8.1 Definitions 193
8.2 Embeddability of L1 195
8.3 Examples 198
8.4 Gd -embeddings of L1 are not narrow 204
8.5 Sign-embeddability of L1 does not imply isomorphic embeddability 208
9 Spaces X for which every operator T. L(Lp,X) is narrow 224
9.1 A characterization using the ranges of vector measures 224
9.2 Every operator from E to c0(G) is narrow 228
9.3 An analog of the Pitt compactness theorem for Lp-spaces 229
9.4 When is every operator from Lp to lr narrow? 230
9.5 l2-strictly singular operators on Lp 239
10 Narrow operators on vector lattices 243
10.1 Two definitions of a narrow operator on vector lattices 244
10.2 AM-compact order-to-norm continuous operators are narrow 248
10.3 T is narrow if and only if |T | is narrow 252
10.4 The Enflo-Starbird function and .-narrow operators 255
10.5 Classical theorems 257
10.6 Pseudonarrow operators are exactly .-narrow operators 260
10.7 Regular narrow operators form a band in the lattice of regular operators 264
10.8 Narrow operators on lattice-normed spaces 266
10.9 l2-strictly singular regular operators are narrow 274
11 Some variants of the notion of narrow operators 279
11.1 Hereditarily narrow operators 280
11.2 Gentle narrow operators on Lp with 1 < p = 2
11.3 C-narrow operators on C(K)-spaces 295
11.4 Narrow operators on L8(µ)-spaces 303
11.5 Narrow 2-homogeneous polynomials 315
12 Open problems 317
Bibliography 321
Index of names 329
Subject index 331

lt;P>"[...] This monograph presents many interesting and deep results revolving around the notion of a narrow operator. The theorems are well motivated and the proofs are very detailed. The authors have edited their text very carefully; also the clear typography makes for a good reading. There are hardly any typos [...] Experts in Banach space theory will appreciate having a copy of this volume at hand."
Dirk Werner, Zentralblatt für Mathematik

"The monograph presents a systematic exposition of the theory of narrow operators and their applications. The proofs are complete. The monograph can be helpful to all beginners as well as advanced researchers in Banach space theory." Mathematical Reviews