Perturbation of minimum attaining operators

Document Type: Original Article


1 Department of Mathematics, IIT Hyderabad, Kandi, Sangareddy, Medak(Dist), Telangana 502285, India

2 IIT Hyderabad


We prove that the minimum attaining property of a bounded linear operator on a Hilbert space $H$ whose minimum modulus lies in the discrete spectrum, is stable under small compact perturbations. We also observe that given a bounded operator with strictly positive essential minimum modulus, the set of compact perturbations which fail to produce a minimum attaining operator is smaller than a nowhere dense set. In fact it is a porous set in the ideal of all compact operators on $H$. Further, we try to extend these stability results to perturbations by all bounded linear operators with small norm and obtain subsequent results.