Absolutely norm attaining paranormal operators

Abstract

A bounded linear operator $T:H_1\to H_2$, where $H_1,H_2$ are Hilbert spaces is said to be norm attaining if there exists a unit vector $x\in H_1$ such that $\Vert Tx\Vert =\Vert T\Vert$. If for any closed subspace M of $H_1$, the restriction $T{|}_M:M\to H_2$ of T to M is norm attaining, then T is called an absolutely norm attaining operator or $\mathcal{AN}$-operator. We prove the following characterization theorem: a positive operator T defined on an infinite dimensional Hilbert space H is an $\mathcal{AN}$-operator if and only if the essential spectrum of T is a single point and $[m(T),m_e(T))$ contains atmost finitely many points. Here $m(T)$ and $m_e(T)$ are the minimum modulus and essential minimum modulus of T, respectively. As a consequence we obtain a sufficient condition under which the $\mathcal{AN}$-property of an operator implies $\mathcal{AN}$-property of its adjoint. We also study the structure of paranormal $\mathcal{AN}$-operators and give a necessary and sufficient condition under which a paranormal $\mathcal{AN}$-operator is normal.