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 l
2
,
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.
Keywords
Compact Operators, Reflexive Banach Spaces, Norm Attainment,
Strong Convergence, Dual Operators, Perturbation Theory, Operator Norms
2020 Mathematics Subject Classification. Primary 46L55; Secondary 44B20
