Application of Norm-Attainable Operators in Functional Analysis

Abstract

Norm-attainable operators are essential in functional analysis, helping solve optimization problems and operator equations in Banach and Hilbert spaces. This study builds on earlier work that focused mainly on Hilbert spaces but gave less attention to other spaces like and the patterns of operator sequences. Using a deductive approach with tools like the Hahn-Banach theorem and spectral theorem, we develop clear conditions for p-norm attainability in spaces and confirm when self-adjoint operators achieve their norm in Hilbert spaces. We extend norm-attainability to iterated operators defined by Banach spaces, show that a sequence {t_n} preserves p-normality, and introduce a new sequence that converges strongly in reflexive spaces. These findings improve methods for maximizing functions and solving operator equations. Our work broadens existing theory by including Banach spaces and new sequence dynamics, providing useful tools for operator algebras. Future studies could explore non-reflexive spaces and sequence convergence speeds