The alternative Dunford-Pettis property in subspaces of operator ideals
Bull. Korean Math. Soc. 2010 Vol. 47, No. 4, 743-750
Published online July 1, 2010
S. Mohammad Moshtaghioun
University of Yazd
Abstract : For several Banach spaces $X$ and $Y$ and operator ideal ${\cal U}$, if ${\cal U}(X,Y)$ denotes the component of operator ideal ${\cal U}$; according to Freedman's definitions, it is shown that a necessary and sufficient condition for a closed subspace ${\cal M}$ of ${\cal U}(X,Y)$ to have the alternative Dunford-Pettis property is that all evaluation operators $\phi_x:{\cal M}\to Y$ and $\psi_{y^*}:{\cal M}\to X^*$ are DP1 operators, where $\phi_x(T)= Tx$ and $\psi_{y^*}(T)= T^*y^*$ for $x\in X$, $y^*\in Y^*$ and $T\in {\cal M}.$
Keywords : Dunford-Pettis property, Schauder decomposition, compact operator, operator ideal
MSC numbers : Primary 47L05, 47L20; Secondary 46B28, 46B99
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