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Observations on some classes of operators on C(K,X)

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Abstract

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 \(T \colon C(K,X)\to Y\) is a strongly bounded operator with representing measure \(m \colon \Sigma \to L(X,Y)\). We show that if \(\hat{T} \colon B(K, X) \to Y\) is its extension, then T is weak Dunford--Pettis (resp.weak* Dunford--Pettis, weak p-convergent, weak* p-convergent) if and only if \(\hat{T}\) has the same property.

We prove that if \(T \colon C(K,X)\to Y\) is strongly bounded limited completely continuous (resp. limited p-convergent), then \(m(A) \colon X\to Y\) is limited completely continuous (resp. limited p-convergent) for each \(A\in \Sigma\). We also prove that the above implications become equivalences when K is a dispersed compact Hausdorff space.

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  1. University of Wisconsin-River Falls, River Falls, Wisconsin, 54022, USA

    I. Ghenciu

  2. University of Pittsburgh, Pittsburgh, PA, 15260, USA

    R. Popescu

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Ghenciu, I., Popescu, R. Observations on some classes of operators on C(K,X). Anal Math (2024). https://doi.org/10.1007/s10476-024-00009-w

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