On the denseness of minimum attaining operators

TL;DR

This paper proves that any densely defined closed operator on Hilbert spaces can be approximated arbitrarily closely by minimum attaining operators through small bounded perturbations, with a special case for bounded below operators.

Contribution

It establishes the denseness of minimum attaining operators in the space of densely defined closed operators on Hilbert spaces, including a rank-one perturbation result for bounded below operators.

Findings

Any densely defined closed operator can be approximated by minimum attaining operators.

For bounded below operators, the approximation can be achieved with a rank-one operator.

The approximation can be made arbitrarily close with a bounded operator of norm less than any given epsilon.

Abstract

Let $H_1,H_2$ be complex Hilbert spaces and $T$ be a densely defined closed linear operator (not necessarily bounded). It is proved that for each $\epsilon>0$, there exists a bounded operator $S$ with $\|S\|\leq \epsilon$ such that $T+S$ is minimum attaining. Further, if $T$ is bounded below, then $S$ can be chosen to be rank one.