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An isomorphic property in spaces of compact operators and some classes of operators on C(K,X)

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Abstract

Let \({K_{w^{*}}(X^{*},Y)}\) denote the set of all w*−w continuous compact operators from X* to Y. We investigate whether the space \({K_{w^{*}}(X^{*},Y)}\) has property RDP * p (\({1\le p < \infty}\)) when X and \({Y}\) have the same property.

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, \({\Sigma}\) is the \({\sigma}\) -algebra of Borel subsets of K, C(K,X) is the Banach space of all continuous X-valued functions (with the supremum norm), and \({T: C(K,X)\to Y}\) is a strongly bounded operator with representing measure \({m: \Sigma \to L(X,Y)}\).

We show that if T is a strongly bounded operator and \({\hat{T}: B(K, X) \to Y}\) is its extension, then T* is p-convergent if and only if \({\hat{T}^{*}}\) is p-convergent, for \({1\le p < \infty}\).

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  1. Mathematics Department, University of Wisconsin–River Falls, Wisconsin, 54022, U.S.A.

    I Ghenciu

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Ghenciu, I. An isomorphic property in spaces of compact operators and some classes of operators on C(K,X). Acta Math. Hungar. 157, 63–79 (2019). https://doi.org/10.1007/s10474-018-0893-9

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