On a subclass of norm attaining operators

Abstract

A bounded linear operator T: H-1 -> H-2, where H-1, H-2 are Hilbert spaces, is said to be norm attaining if there exists a unit vector x is an element of H-1 such that parallel to Tx parallel to = parallel to T parallel to and absolutely norm attaining (or AN-operator) if T vertical bar M: M -> H-2 is norm attaining for every closed subspace M of H-1. We prove a structure theorem for positive operators in beta(H) := {T is an element of B(H) : T vertical bar(M) : M -> M is norm attaining for all M is an element of R-T}, where R-T is the set of all reducing subspaces of T. We also compare our results with those of absolutely norm attaining operators. Later, we characterize all operators in this new class.