When strict singularity of operators coincides with weak compactness
Let \(X\) stands for one of the spaces (1) \(C(K)\); (2) \(A(\mathbb D)\); (3) \(X\) is a subspace of \(C(K)\) with reflexive annihilator; (4) \(X\) is a subspace of the Mores-Transue space \(M^{\psi _q}(\Omega ,\mu )\) with \(q>2\), on a probability space. The main result of the paper isNEWLINENEWLINE {Theorem 2.9.} Let \(X\) be one of the spaces listed above and \(T\) be any bounded operator from \(X\) to a Banach space \(Y\). Then the following assertions are equivalent:NEWLINENEWLINE (i) \(T\) is a finitely strictly singular operator;NEWLINENEWLINE (ii) \(T\) is a strictly singular operator;NEWLINENEWLINE (iii) \(T\) is a weakly compact operator.NEWLINENEWLINE Moreover, for the spaces (1), (2) and (3) in the list above, these notions also coincide with the notion of complete continuity.
scientific article; zbMATH DE number 6045973