$L_{\phi }$-Spaces and some Related Sequence Spaces

Abstract

In view of closed graph theorems in case of maps defined by operator-valued matrices $L_{\phi }$-spaces were recently introduced by two of the present authors as a generalization of separable $FK(X)$-spaces. In this paper we study the class of $L_{\phi }$-spaces and a few closely related classes of sequence spaces. It is shown that an analogue of Kalton’s closed graph theorem holds for matrix mappings if we consider $L_{\phi }$-spaces as range spaces, and paralleling a result of Qiu we prove that the class of $L_{\phi }$-spaces is the best-possible choice here. As a consequence we show that for any $L_{\phi }$-space $E$ every matrix domain $E_A$ is again an $L_{\phi }$-space.