The Kalton and Rosenthal type decomposition of operators in Köthe-Bochner spaces
type: Item
instance of: scholarly article
title: The Kalton and Rosenthal type decomposition of operators in Köthe-Bochner spaces
zbMATH Open document ID: 1496.47065
published in: Journal of Mathematical Analysis and Applications
publication date: 2021-04-23T00:00:00Z
review text: The paper under review is partly motivated by a classical result of \textit{N. J. Kalton} [Indiana Univ. Math. J. 27, 353--381 (1978; Zbl 0403.46032)] which in particular implies that every operator \(T:L_1(\mu)\rightarrow L_1(\mu)\) admits a decomposition \(T=T_a+T_d\) where \(T_a\) is a pseudo-integral operator with respect to a family of absolutely continuous measures and \(T_d\) is an atomic operator. This decomposition is here extended to a certain class of operators (dominated operators) from a certain family of lattice-normed spaces (decomposable spaces) into order continuous Banach lattices. Here, a lattice-normed space is a triple \((E,V,p)\), where \(E\) is a vector space, \(V\) is an Archimedean vector lattice, and \(p:E\rightarrow V\) is a map satisfying: (1) \(p(x)\geq0\), with \(p(x)=0\) if and only if \(x=0\); (2) \(p(x_1+x_2)\leq p(x_1)+p(x_2)\); (3) \(p(\lambda x)=|\lambda|p(x)\). In this setting, several interesting results are obtained concerning the class of disjointness preserving and narrow operators. Moreover, the results are in particular applied to the situation of operators on Köthe-Bochner spaces. We refer the interested reader to the paper for details.
reviewed by: P. Tradacete
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 7337428
zbMATH Keywords: narrow operator
zbMATH Keywords: dominated operator
zbMATH Keywords: diffuse operator
zbMATH Keywords: disjointness preserving operator
zbMATH Keywords: Kalton-Rosenthal decomposition
zbMATH Keywords: Köthe-Bochner space
author: M. Pliev
author: Fedor Sukochev
MaRDI profile type: Publication
full work available at URL: j.jmaa.2021.125142
OpenAlex ID: W3137987711
cites work: Completely additive and \(C\)-compact operators in lattice-normed spaces
cites work: On the sum of narrow orthogonally additive operators
cites work: Dominated operators from lattice-normed spaces to sequence Banach lattices
cites work: Dominated orthogonally additive operators in lattice-normed spaces
cites work: On two definitions of a narrow operator on Köthe-Bochner spaces
cites work: Q4781239
cites work: Compact-like operators in lattice-normed spaces
cites work: Q4746343
cites work: Q5386174
cites work: Linear biseparating maps between spaces of vector-valued differentiable functions and automatic continuity
cites work: On orthogonally additive operators in Köthe-Bochner spaces
cites work: Q5322320
cites work: Multiplication operators in Köthe-Bochner spaces
cites work: Maximal factorization of operators acting in Köthe-Bochner spaces
cites work: Multiplication operators on vector-valued function spaces
cites work: Biseparating linear maps between continuous vector-valued function spaces
cites work: On sums of narrow and compact operators
cites work: An alternative proof of a Radon-Nikodým theorem for lattice homomorphisms
cites work: Q3977662
cites work: Q4216778
cites work: Characterizations of Disjointness preserving operators on vector-valued function spaces
cites work: Q4188290
cites work: Q4520849
cites work: Q4188284
cites work: Q4452867
cites work: A lattice approach to narrow operators
cites work: On sums of strictly narrow operators acting from a Riesz space to a Banach space
cites work: Continuous linear operators on Orlicz-Bochner spaces
cites work: Narrow operators on lattice-normed spaces
cites work: Narrow orthogonally additive operators in lattice-normed spaces
cites work: Q3202684
cites work: Narrow operators (a survey)
cites work: Narrow operators on function spaces and vector lattices
cites work: Narrow and \(C\)-compact orthogonally additive operators in lattice-normed spaces
cites work: Embeddings of 𝐿¹ in 𝐿¹
cites work: On the Representation of Order Continuous Operators by Random Measures
DOI: 10.1016/J.JMAA.2021.125142
Recommended article: Dominated operators. III
Recommended article: Dominated operators from lattice-normed spaces to sequence Banach lattices
Recommended article: Narrow operators on lattice-normed spaces
Recommended article: On orthogonally additive operators in Köthe-Bochner spaces
Recommended article: Q4520849
Recommended article: Narrow Operators on Lattice-normed Spaces and Vector Measures
Recommended article: Domination properties of lattice homomorphisms
Recommended article: Decomposition theorems for disjointness preserving operators
Recommended article: Domination by positive narrow operators
Recommended article: A note on compact Lamperti operators
description: scientific article; zbMATH DE number 7337428