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 $ε>0$, there exists a bounded operator $S$ with $\|S\|\leq ε$ such that $T+S$ is minimum attaining. Further, if $T$ is bounded below, then $S$ can be chosen to be rank one.

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