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‖ ≤ ε such that T + S is minimum attaining. Further, if T is bounded below, that is if there exists m > 0 such that ‖Tx‖ ≥ m‖x‖ for every x in the domain of T, then S can be chosen to be rank one.