Norm attainment for compact operators on reflexive Banach spaces

Abstract

This paper explores the norm attainment of compact operators on reflexive Banach spaces, emphasizing the interplay between geometric and structural properties. Key results include characterizations of norm attainment through sequences in the unit sphere, uniqueness of norm attainment in strictly convex spaces, and stability of norm attainment under compact perturbations. Notable findings include that self-adjoint compact operators on spaces with symmetric unit balls attain their norms at symmetric points, and that dual operators also attain their norms under specific conditions. Examples on l², C[0,1], and L²([0,1]) demonstrate the practical implications of these theoretical results. This work deepens the understanding of compact operators and their significance in functional analysis, spectral theory, and optimization.