Affiliation:
1. Department of Mathematics, Faculty of Sciences, University of Gabes, Erriadh City Zrig, Gabes, Tunisia
2. Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Tunisia
Abstract
A pair (u, ?) in X ? X?, where X is an infinite dimensional Banach space and
X? its topological dual space, induces in a natural way two multiplication
operators L?,u and R?,u on the Banach space L(X), defined by L?,u (T) (x)
= ? (T(x)) u, and R?,u (T) (x) = ?(x)T(u), for all T in L(X) and x in X.
In this paper, we present necessary and sufficient conditions for the
compactness, demicompactness, stongly demicompactess, power compactness and
Riesz property of this family of operators. We also establish sufficient
conditions for the quasi-compactness and weak compactness of these
operators. Finally, we show that the Dunford-Pettis property fails for the
Banach space L(X) whenever either X or L(X) is reflexive.
Publisher
National Library of Serbia
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