Norm Attainability of Compact Operators: Spectral, Geometric, and Perturbation Insights

This paper presents a comprehensive investigation into the norm attainability of compact operators on infinitedimensional Hilbert spaces, offering novel spectral and geometric insights. We establish necessary and sufficient conditions for norm attainment in terms of the spectral structure of the operator, demonstrating that a compact operator T attains its norm if and only if ∥T∥ is an eigenvalue of |T| = \(\sqrt{T*T}\) with a corresponding eigenvector of unit norm. Our results extend to Schatten class operators, highlighting the interplay between norm attainment, singular values, and maximizing sequences. Furthermore, we explore norm attainment under perturbations, revealing stability conditions and spectral dominance properties that ensure preservation of normattaining behavior. These findings contribute to a deeper understanding of operator theory with applications in quantum mechanics, signal processing, and functional analysis.