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Complemented Subspaces of Linear Bounded Operators
Published online by Cambridge University Press: 20 November 2018
Abstract
We study the complementation of the space $W\left( X,Y \right)$ of weakly compact operators, the space
$K\left( X,Y \right)$ of compact operators, the space
$U\left( X,Y \right)$ of unconditionally converging operators, and the space
$CC\left( X,Y \right)$ of completely continuous operators in the space
$L\left( X,Y \right)$ of bounded linear operators from
$X$ to
$Y$. Feder proved that if
$X$ is infinite-dimensional and
${{c}_{0}}\,\to \,Y$, then
$K\left( X,Y \right)$ is uncomplemented in
$L\left( X,Y \right)$. Emmanuele and John showed that if
${{c}_{0}}\,\to \,K(X,\,Y)$, then
$K\left( X,Y \right)$ is uncomplemented in
$L\left( X,Y \right)$. Bator and Lewis showed that if
$X$ is not a Grothendieck space and
${{c}_{0}}\,\to \,Y$, then
$W\left( X,Y \right)$ is uncomplemented in
$L\left( X,Y \right)$. In this paper, classical results of Kalton and separably determined operator ideals with property
$\left( * \right)$ are used to obtain complementation results that yield these theorems as corollaries.
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- Copyright © Canadian Mathematical Society 2012
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