On Characterization of Norm Attaining Operators in Fr´echet Spaces

Abstract

In this paper, the norm attaining operators in Fr´echet spaces are considered. These operators are characterized based on their density, normality, linearity and compactness. It is shown that the image is dense for a normal and injective operator in a Fr´echet space, as well as its inverse given that the operator is self-adjoint. A norm attaining operator in a Fr´echet space is also shown to be normal if its adjoint also attains its norm in the Fr´echet space, and the condition under which the norm attainability and the normality of an operator in a Fr´echet space coincides is given. Furthermore, a norm attaining operator between Fr´echet spaces is linear and bounded as well as its inverse. If a norm attaining, normal and dense operator is of finite rank, then it is compact. The study of norm attaining operators is applicable in algorithm concentration as seen in describing sphere packing.