NORM ATTAINING OPERATORS IN FRECHET SPACES

Abstract

Norm attaining operators have been investigated in particular with regard to their characterization. Numerous studies have emphasized the denseness property in the characterization of norm attaining operators, primarily in Banach and Hilbert spaces. Similarly, Fr´echet spaces have been studied and various findings have been obtained. Due to the fundamental fact that the concept of Fr´echet spaces generalizes both the notions of Banach spaces and Hilbert spaces, it was of significant interest to investigate the characterization of norm attaining operators in these spaces. The main objective of this study was to characterize norm attaining operators in Fr´echet spaces. The specific objectives were; to determine classes of norm attaining operators in Fr´echet spaces, to characterize the classes of norm attaining operators in Fr´echet spaces via density and to determine a classification of the classes of norm attaining operators with respect to linearity, normality and compact ness. The methods of this study included the use of Approximation Prop erty, Bishop-Phelps property and Radon-Nikodym property. The findings demonstrate that, among the classes of norm attaining operators in Fr´echet spaces that have been determined, the classes of norm attaining operators of finite rank and norm attaining normal operators were taken into account in consideration of the two conditions, namely the existence of a finite rank operator and the existence of a finite dimensional space. The results further demonstrate that the image is dense for a normal and injective operator in a Fr´echet space. Additionally, it is proven that a dense operator’s adjoint in a Fr´echet space is dense if the operator is self-adjoint. Furthermore, it is demonstrated that there are other properties besides density, which include normality, linearity and compactness, that can be used to characterize norm attaining operators in Fr´echet spaces. A generalized class of norm attaining operators is also determined with respect to the aforementioned properties. The findings of this research are essential in Operator Theory specifically incomplete classification and applications of operators in Fr´echet spaces. In addition, the results are applicable in algorithms concentration as seen in describing sphere packing. In particular, a dense and polydisperse sphere packing algorithm is of great interest for engineering problems since very large packings can be built in a few minutes as facilitated by the concept of norm attaining operators.