Stability of \({\cal A}{\cal N}\)-Operators under Functional Calculus

Abstract

In this note we discuss absolutely norm attaining property (\({\cal A}{\cal N}\)-property in short) of the Jordan product and Lie-bracket. We propose a functional calculus for positive absolutely norm attaining operators and discuss the stability of the \({\cal A}{\cal N}\)-property under the functional calculus. As a consequence we discuss the operator mean of positive \({\cal A}{\cal N}\)-operators.