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A note on Dunford-Pettis like properties and complemented spaces of operators. (English) Zbl 1463.46033

Summary: Equivalent formulations of the Dunford-Pettis property of order \(p\) (\(\text{DPP}_p\)), \(1<p<\infty\), are studied. Let \(L(X,Y)\), \(W(X,Y)\), \(K(X,Y)\), \(U(X,Y)\), and \(C_p(X,Y)\) denote respectively the sets of all bounded linear, weakly compact, compact, unconditionally converging, and \(p\)-convergent operators from \(X\) to \(Y\). Classical results of Kalton are used to study the complementability of the spaces \(W(X,Y)\) and \(K(X,Y)\) in the space \(C_p(X,Y)\), and of \(C_p(X,Y)\) in \(U(X,Y)\) and \(L(X,Y)\).

MSC:

46B28 Spaces of operators; tensor products; approximation properties
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References:

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