Absolutely norm attaining paranormal operators

A bounded linear operator T:H1→H2, where H1,H2 are Hilbert spaces is said to be norm attaining if there exists a unit vector x∈H1 such that ‖Tx‖=‖T‖. If for any closed subspace M of H1, the restriction T|M:M→H2 of T to M is norm attaining, then T is called an absolutely norm attaining operator or AN-operator. We prove the following characterization theorem: a positive operator T defined on an infinite dimensional Hilbert space H is an AN-operator if and only if the essential spectrum of T is a single point and [m(T),me(T)) contains atmost finitely many points. Here m(T) and me(T) are the minimum modulus and essential minimum modulus of T, respectively. As a consequence we obtain a sufficient condition under which the AN-property of an operator implies AN-property of its adjoint. We also study the structure of paranormal AN-operators and give a necessary and sufficient condition under which a paranormal AN-operator is normal.