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Dunford-Pettis-like properties of continuous vector function spaces. (English) Zbl 0807.46033

Summary: The structure of some operator ideals \({\mathfrak A}\) defined on continuous function spaces is studied. Conditions are considered under which “\(T\in {\mathfrak A}\)” and “the representing measure of \(T\) takes values in \({\mathfrak A}\)” are equivalent for the scales of \(p\)-converging \((C_ p)\) and weakly-\(p\)-compact \((W_ p)\) operators. The scale \(C_ p\) is intermediate between the ideals \(C_ 1={\mathfrak U}\) (unconditionally summing operators), and \(C_ \infty={\mathfrak B}\) (completely continuous operators), which have been studied by several authors (Bombal, Cembranos, Rodríguez-Salinas, Saab). The dual scale \(W_ p\) is intermediate between the ideals \(\mathfrak K\) (compact operators) and \(W_ \infty =W\) (weakly compact operators), and the results presented have a close connection with those of Diestel, Núñez and Seifert.

MSC:

46E40 Spaces of vector- and operator-valued functions
47L20 Operator ideals
46B28 Spaces of operators; tensor products; approximation properties
46E15 Banach spaces of continuous, differentiable or analytic functions
46B25 Classical Banach spaces in the general theory
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