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Zeitschrift für Analysis und ihre Anwendungen


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Volume 13, Issue 3, 1994, pp. 377–385
DOI: 10.4171/ZAA/507

Published online: 1994-09-30

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

Johann Boos[1], Karl-Goswin Grosse-Erdmann[2] and T. Leiger[3]

(1) Fernuniversität-GHS, Hagen, Germany
(2) Fernuniversität-GHS, Hagen, Germany
(3) Tartu University, Estonia

In view of closed graph theorems in case of maps defined by operator-valued matrices $L_{\varphi}$-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 $L_{\varphi}$-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 $L_{\varphi}$-spaces as range spaces, and paralleling a result of Qiu we prove that the class of $L_{\varphi}$-spaces is the best-possible choice here. As a consequence we show that for any $L_{\varphi}$-space $E$ every matrix domain $E_A$ is again an $L_{\varphi}$-space.

Keywords: Matrix mappings, closed graph theorems, $L_{\varphi}$-spaces, $L_r$-spaces

Boos Johann, Grosse-Erdmann Karl-Goswin, Leiger T.: $L_{\varphi}$-Spaces and some Related Sequence Spaces. Z. Anal. Anwend. 13 (1994), 377-385. doi: 10.4171/ZAA/507