The Kalton and Rosenthal type decomposition of operators in Köthe-Bochner spaces
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.
scientific article; zbMATH DE number 7337428
scientific article