M-ideals of compact operators and Norm attaining operators

Abstract

We investigate M-ideals of compact operators and two distinct properties in norm-attaining operator theory related with M-ideals of compact operators called the weak maximizing property and the compact perturbation property. For Banach spaces $X$ and $Y$, it is previously known that if $\mathcal{K}(X,Y)$ is an M-ideal or $(X,Y)$ has the weak maximizing property, then $(X,Y)$ has the adjoint compact perturbation property. We see that their converses are not true, and the condition that $\mathcal{K}(X,Y)$ is an M-ideal does not imply the weak maximizing property, nor vice versa. Nevertheless, we see that all of these are closely related to property $(M)$, and as a consequence, we show that if $\mathcal{K}(\ell_p,Y)$ $(1<p<\infty)$ is an M-ideal, then $(\ell_p,Y)$ has the weak maximizing property. We also prove that $(\ell_1,\ell_1)$ does not have the adjoint compact perturbation property, and neither does $(\ell_1,Y)$ for an infinite dimensional Banach space $Y$ without an isomorphic copy of $\ell_1$ if $Y$ does not have the local diameter 2 property. As a consequence, we show that if $Y$ is an infinite dimensional Banach space such that $\mathcal{L}(\ell_1,Y)$ is an M-ideal, then it has the local diameter 2 property. Furthermore, we also studied various geometric properties of Banach spaces such as the Opial property with moduli of asymptotic uniform smoothness and uniform convexity.