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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