On the closure of absolutely norm attaining operators

Abstract

ABSTRACTLet H1 and H2 be complex Hilbert spaces and T:H1→H2 be a bounded linear operator. We say T is norm attaining if there exists x∈H1 with ‖x‖=1 such that ‖Tx‖=‖T‖. If for every non-zero closed subspace M of H1, the restriction T|M:M→H2 is norm attaining, then T is called an absolutely norm attaining operator or AN-operator. If we replace the norm of the operator by the minimum modulus m(T)=inf{‖Tx‖:x∈H1,‖x‖=1} in the above definitions, then T is called a minimum attaining and an absolutely minimum attaining operator or AM-operator, respectively. In this article, we discuss the operator norm closure of AN-operators. We completely characterize operators in this closure and study several important properties. We mainly give a spectral characterization of positive operators in this class and give a representation when the operator is normal. Later, we also study the analogous properties for AM-operators and prove that the closure of AM-operators is the same as the closure of AN-operators. Consequently, we prove similar results for operators in the norm closure of AM-operators.