Hereditary properties of certain ideals of compact operators
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Abstract
Let $X$ be a Banach space and $Z$ a closed subspace of a Banach space $Y.$ Denote by $\CL(X,Y)$ the space of all bounded linear operators from $X$ to $Y$ and by $\CK(X,Y)$ its subspace of compact linear operators. Using Hahn-Banach extension operators corresponding to ideal projections, we prove that if either $X^{**}$ or $Y^*$ has the Radon-Nikod$\acute{y}$m property and $\CK(X,Y)$ is an $M$-ideal (resp. an $HB$-subspace) in $\CL(X,Y),$ then $\CK(X,Z)$ is also an $M$-ideal (resp. $HB$-subspace) in $\CL(X,Z).$ If $\CK(X,Y)$ has property $SU$ instead of being an $M$-ideal in $\CL(X,Y)$ in the above, then $\CK(X,Z)$ also has property $SU$ in $\CL(X,Z).$ If $X$ is a Banach space such that $X^*$ has the metric compact approximation property with adjoint operators, then $M$-ideal (resp. $HB$-subspace) property of $\CK(X,Y)$ in $\CL(X,Y)$ is inherited to $\CK(X,Z)$ in $\CL(X,Z).$
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