Abstract:
This paper investigates the geometry and norm-attainability of operators within various operator
ideals, with a particular focus on the role of singular values and compactness. We explore the behavior of
norm-attainable operators in the context of classical operator ideals, such as trace-class and Hilbert-Schmidt
operators, and examine how their geometric and algebraic properties are influenced by membership in these
ideals. A key result of this study is the connection between the singular values of trace-class operators
and their operator norm, establishing a foundational relationship for understanding norm-attainment.
Additionally, we explore the conditions under which weakly compact and compact operators can attain their
operator norm, providing further insights into the structural properties that govern norm-attainability in
operator theory. The findings contribute to a deeper understanding of the interplay between operator ideals
and norm-attainability, with potential applications in functional analysis and related fields.
