Uniform convergence of operators on \(L^{\infty}\) and similar spaces
Let E be a Banach space. It is said to have the Dunford-Pettis property if every weakly compact operator from E into any Banach space maps weakly compact sets into norm compact sets. It is called a Grothendieck space if every weak* convergent sequence in the dual \(E^*\) converges also for the weak topology of \(E^*.\) The paper is concerned with the uniform convergence of a sequence of bounded linear operators on a Grothendieck space with the Dunford-Pettis property. Sequences which are relatively compact for the strong or the weak operator topology are considered and their convergence in one of these topologies is discussed. It is shown that uniform convergence is possible under some additional hypotheses of algebraic, topological or spectral theoretic nature. The discussion is classified into (1) operators on C(X), (2) semi-groups of operators, (3) ergodic theorems, and (4) Tauberian theorems. Among the many interesting results is one which solves a problem of \textit{A. Kishimoto} and \textit{D. W. Robinson} [J. Austr. Math. Soc., Ser. A 31, 59-76 (1981; Zbl 0491.47021), p. 75], namely, any strongly continuous one-parametric semi-group of bounded linear operators on a Grothendieck space with Dunford-Pettis property is necessarily uniformly continuous.
uniform convergence of a sequence of bounded linear operators on a Grothendieck space with the Dunford-Pettis property
scientific article; zbMATH DE number 4011258
scientific article