When strict singularity of operators coincides with weak compactness. (English) Zbl 1261.47028
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 is
{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:
(i) \(T\) is a finitely strictly singular operator;
(ii) \(T\) is a strictly singular operator;
(iii) \(T\) is a weakly compact operator.
Moreover, for the spaces (1), (2) and (3) in the list above, these notions also coincide with the notion of complete continuity.
{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:
(i) \(T\) is a finitely strictly singular operator;
(ii) \(T\) is a strictly singular operator;
(iii) \(T\) is a weakly compact operator.
Moreover, for the spaces (1), (2) and (3) in the list above, these notions also coincide with the notion of complete continuity.
Reviewer: Dumitrŭ Popa (Constanţa)
MSC:
| 47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |
| 47B38 | Linear operators on function spaces (general) |
| 47B07 | Linear operators defined by compactness properties |
