Journal of Operator Theory

Volume 49, Issue 1, Winter 2003 pp. 143-151.

Weak sequential convergence in the dual of operator ideals

Authors: S.M. Moshtaghioun (1) and J. Zafarani (2)
Author institution: (1) Department of Mathematics, University of Isfahan, Isfahan, 81745-163, Iran
(2) Department of Mathematics, University of Isfahan, Isfahan, 81745-163, Iran

Summary: By giving some necessary and sufficient conditions for the dual of operator subspaces to have the Schur property, we improve the results of Brown, \"Ulger and Saksman-Tylli in the Banach space setting. In particular, under some conditions on Banach spaces

$X$ and

$Y$ , we show that for a subspace

${\cal M}$ of operator ideal

${\cal U}(X,Y)$ ,

${\cal M}^*$ has the Schur property iff all point evaluations

${\cal M}_1(x)= \{Tx: T\in {\cal M}_1 \}$ and

$\widetilde {\cal M}_1(y^*)= \{T^*y^*: T\in {\cal M}_1 \}$ are relatively norm compact, where

$x\in X$ ,

$y^*\in Y^*$ and

${\cal M}_1$ is the closed unit ball of

${\cal M}$ .

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