A BISHOP-PHELPS-BOLLOBAS TYPE PROPERTY FOR MINIMUM ATTAINING OPERATORS

In this article, we study the Bishop-Phelps-Bollobas type theorem for minimum attaining operators. More explicitly, if we consider a bounded linear operator T on a Hilbert space H and a unit vector x(0) is an element of H such that parallel to Tx(0)parallel to is very close to the minimum modulus of T, then T and x0 are simultaneously approximated by a minimum attaining operator S on H and a unit vector y is an element of H for which parallel to Sy parallel to is equal to the minimum modulus of S. Further, we extend this result to a more general class of densely defined closed operators (need not be bounded) in Hilbert space. As a consequence, we get the denseness of the set of minimum attaining operators in the class of densely defined closed operators with respect to the gap metric.