Ball proximinality of 𝑀-ideals of compact operators

Abstract

In this article, we prove the proximinality of closed unit ball of M 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 K ( ℓ 1 ) \mathcal {K}(\ell _1) , the space of compact operators on ℓ 1 \ell _1 , is ball proximinal in B ( ℓ 1 ) \mathcal {B}(\ell _1) , the space of bounded operators on ℓ 1 \ell _1 , even though K ( ℓ 1 ) \mathcal {K}(\ell _1) is not an M M -ideal in B ( ℓ 1 ) \mathcal {B}(\ell _1) . Moreover, we prove the ball proximinality of M M -embedded spaces in their biduals.