On Characterization of Norm Attaining Operators in Frechet Spaces
Vashti Bernice *
Department of Mathematics and Statistics, Kaimosi Friends University, P.O. Box 385-50309, Kaimosi, Kenya.
Amos Wanjara
Department of Mathematics and Statistics, Kaimosi Friends University, P.O. Box 385-50309, Kaimosi, Kenya.
Michael Ojiema
Department of Mathematics, Masinde Muliro University of Science and Technology, P.O. Box 190-50100, Kakamega, Kenya.
*Author to whom correspondence should be addressed.
Abstract
In this paper, the norm attaining operators in Frechet 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 Frechet space, as well as its inverse given that the operator is self-adjoint. A norm attaining operator in a Frechet space is also shown to be normal if its adjoint also attains its norm in the Frechet space, and the condition under which the norm attainability and the normality of an operator in a Frechet space coincides is given. Furthermore, a norm attaining operator between Frechet 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.
Keywords: Norm attaining operator, Frechet space, Characterization