On the denseness of minimum attaining operators
  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. 
S. H. Kulkarni, G. Ramesh
math.FA