Ball proximinality of $M$-ideals of compact operators

Abstract:

In this article, we prove the proximinality of closed unit ball of $M$-ideals of compact operators on Banach spaces. We show that every positive (self-adjoint) operator on a Hilbert space has a positive (self-adjoint) compact approximant from the closed unit ball of space of compact operators. We also show that $\mathcal {K}(\ell _1)$, the space of compact operators on $\ell _1$, is ball proximinal in $\mathcal {B}(\ell _1)$, the space of bounded operators on $\ell _1$, even though $\mathcal {K}(\ell _1)$ is not an $M$-ideal in $\mathcal {B}(\ell _1)$. Moreover, we prove the ball proximinality of $M$-embedded spaces in their biduals.