Perturbation of minimum attaining operators

Abstract

We prove that the minimum attaining property of a bounded linear operator on a Hilbert space $H$ whose minimum modulus lies in the discrete spectrum, is stable under small compact perturbations. We also observe that given a bounded operator with strictly positive essential minimum modulus, the set of compact perturbations which fail to produce a minimum attaining operator is smaller than a nowhere dense set. In fact, it is a porous set in the ideal of all compact operators on $H$. Further, we try to extend these stability results to perturbations by all bounded linear operators with small norm and obtain subsequent results.