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Characterization of some classes of operators on spaces of vector-valued continuous functions

Bombal Gordón, Fernando and Cembranos Diaz, Pilar (1985) Characterization of some classes of operators on spaces of vector-valued continuous functions. Mathematical Proceedings of the Cambridge Philosophical Society , 97 (1). pp. 137-146. ISSN 0305-0041

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Abstract

Let $K$K be a compact Hausdorff space and $E$E, $F$F Banach spaces with $L(E,F)$L(E,F) the space of bounded linear operators from $E$E into $F$F. If $C(K,E)$C(K,E) is the space of all continuous functions from $K$K into $E$E equipped with the sup-norm, then every operator $T\in L(C(K,E),F)$T∈L(C(K,E),F) has a representing measure $m$m of bounded semivariation on the Borel sets of $K$K with values in $L(E,F'')$L(E,F′′) such that $TF=\int_Kf\,dm$TF=∫Kfdm. If $T$T is a weakly compact operator, then $m$m has values in $L(E,F)$L(E,F), $m(E)$m(E) is weakly compact for each Borel set $E$E, and the semivariation of $m$m is continuous at $\varphi$φ. It is known that the converse of this statement does not hold in general, but does hold under additional assumptions. In particular, the authors show that the converse holds if $K$K is a dispersed space. They also show that, in a certain sense, the assumption that $K$K is a dispersed space is necessary; that is, if the converse of the statement above holds for every pair of Banach spaces $E,F$E,F then $K$K must be a dispersed space. A similar result holds for the class of unconditionally converging, Dunford-Pettis or Dieudonne operators.


Item Type:Article
Uncontrolled Keywords:Spaces of vector-valued continuous functions; Class of weakly compact; Dunford-Pettis; Dieudonn´e or unconditionally converging operators; Representing measure; Semi-variation
Subjects:Sciences > Mathematics > Mathematical analysis
ID Code:15155
Deposited On:09 May 2012 09:33
Last Modified:06 Feb 2014 10:17

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