Non-complemented Spaces of Operators, Vector Measures, and $c_o$

Canadian mathematical bulletin

Polly Schulle¹  Paul Lewis² 

[1] Department of Mathematics, Richland College, Dallas, TX 75243-2199 USA;Department of Mathematics, University of North Texas, Denton, TX 76203-1430 USA

关键词: spaces of operators;    compact operators;    complemented subspaces;    $w^*-w$-compact operators;   

DOI  :  10.4153/CMB-2011-084-0

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

The Banach spaces $L(X, Y)$, $K(X, Y)$, $L_{w^*}(X^*, Y)$, and$K_{w^*}(X^*, Y)$ are studied to determine when they contain theclassical Banach spaces $c_o$ or $ell_infty$. The complementation ofthe Banach space $K(X, Y)$ in $L(X, Y)$ is discussed as well as whatimpact this complementation has on the embedding of $c_o$ or$ell_infty$ in $K(X, Y)$ or $L(X, Y)$. Results of Kalton, Feder, andEmmanuele concerning the complementation of $K(X, Y)$ in $L(X, Y)$ aregeneralized. Results concerning the complementation of the Banachspace $K_{w^*}(X^*, Y)$ in $L_{w^*}(X^*, Y)$ are also explored as wellas how that complementation affects the embedding of $c_o$ or$ell_infty$ in $K_{w^*}(X^*, Y)$ or $L_{w^*}(X^*, Y)$. The $ell_p$spaces for $1 = p < infty$ are studied to determine when the space ofcompact operators from one $ell_p$ space to another contains$c_o$. The paper contains a new result which classifies these spacesof operators. A new result using vector measures is given toprovide more efficient proofs of theorems by Kalton, Feder, Emmanuele,Emmanuele and John, and Bator and Lewis.

【 授权许可】

Unknown