On the Closure of Absolutely Norm Attaining Operators

Let H-1 and H-2 be complex Hilbert spaces and T : H-1 -> H-2 be a bounded linear operator. We say T is norm attaining if there exists x is an element of H-1 with parallel to x parallel to = 1 such that parallel to Tx parallel to = parallel to T parallel to. If for every non-zero closed subspaceMof H-1, the restriction T|(M) : M -> H-2 is norm attaining, then T is called an absolutely norm attaining operator or ANoperator. If we replace the norm of the operator by the minimum modulus m(T) = inf {parallel to Tx parallel to : x is an element of H-1, parallel to x parallel to = 1} in the above definitions, then T is called a minimum attaining and an absolutely minimumattaining operator orAM-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. Wemainly 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 AMoperators and prove that the closure ofAM-operators is the same as the closure ofAN-operators. Consequently, we prove similar results for operators in the norm closure ofAM-operators.