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

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.