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Author Popov, Mykhaĭlo Mykhaĭlovych.

Title Narrow operators on function spaces and vector lattices / Mikhail Popov, Beata Randrianantoanina
Published Berlin : De Gruyter, 2012
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Description 1 online resource (336 pages)
Series De Gruyter Studies in Mathematics ; v.45
De Gruyter studies in mathematics.
Contents Preface; 1 Introduction and preliminaries; 1.1 Background information; 1.2 Terminology and notation; 1.3 Narrow operators on function spaces; 1.4 Homogeneous measure spaces and Maharam's theorem; 1.5 Necessary information on vector lattices; 1.6 Kalton's and Rosenthal's representation theorems for operators on L1; 2 Each "small" operator is narrow; 2.1 AM-compact and Dunford-Pettis operators are narrow; 2.2 "Large" subspaces are exactly strictly rich; 2.3 Operators with "small" ranges are narrow; 2.4 Narrow operators are compact on a suitable subspace
3 Applications to nonlocally convex spaces3.1 Nonexistence of nonzero narrow operators; 3.2 The separable quotient space problem; 3.3 Isomorphic classification of strongly nonconvex Köthe F-spaces; 4 Noncompact narrow operators; 4.1 Conditional expectation operators; 4.2 A narrow projection from E onto a subspace isometric to E; 4.3 A characterization of narrow conditional expectations; 5 Ideal properties, conjugates, spectrum and numerical radii; 5.1 Ideal properties of narrow operators and stability of rich subspaces; 5.2 Conjugates of narrow operators need not be narrow
5.3 Spectrum of a narrow operator5.4 Numerical radii of narrow operators on Lp (μ)-spaces; 6 Daugavet-type properties of Lebesgue and Lorentz spaces; 6.1 A generalization of the DP for L1 to "small" into isomorphisms; 6.2 Pseudo-Daugavet property for narrow operators on Lp, p ≠ 2; 6.3 A pseudo-Daugavet property for narrow projections in Lorentz spaces; 6.4 Near isometric classification of Lp (μ)-spaces for 1 ≤ p<∞, p ≠ 2; 7 Strict singularity versus narrowness; 7.1 Bourgain-Rosenthal's theorem on l1 -strictly singular operators; 7.2 Rosenthal's characterization of narrow operators on L1
7.3 Johnson-Maurey-Schechtman-Tzafriri's theorem7.4 An application to almost isometric copies of L1; 7.5 An application to complemented subspaces of Lp; 7.6 The Daugavet property for rich subspaces of L1; 8 Weak embeddings of L1; 8.1 Definitions; 8.2 Embeddability of L1; 8.3 Examples; 8.4 Gδ -embeddings of L1 are not narrow; 8.5 Sign-embeddability of L1 does not imply isomorphic embeddability; 9 Spaces X for which every operator T∊ L(Lp,X) is narrow; 9.1 A characterization using the ranges of vector measures; 9.2 Every operator from E to c0(Γ) is narrow
9.3 An analog of the Pitt compactness theorem for Lp-spaces9.4 When is every operator from Lp to lr narrow?; 9.5 l2-strictly singular operators on Lp; 10 Narrow operators on vector lattices; 10.1 Two definitions of a narrow operator on vector lattices; 10.2 AM-compact order-to-norm continuous operators are narrow; 10.3 T is narrow if and only if
Summary Narrow operators are those operators defined on function spaces which are ""small'' at signs, i.e. at {-1,0,1}-valued functions. Numerous works and research papers exist on these, but no coherent monograph yet to place them in context. This book gives comprehensive treatment of narrow operators. It starts with basics and then systematically builds up the case. It also covers geometrical applications and Gaussian embeddings
Notes 10.8 Narrow operators on lattice-normed spaces
Description based upon print version of record
Subject Function spaces.
Narrow operators.
Riesz spaces.
Form Electronic book
Author Randrianantoanina, Beata.
ISBN 3110263343