Density and dentability in norm-attainable classes

Abstract

We establish the norm-denseness of the norm-attainable class NA(H) in the Banach algebra B(H), which consists of all bounded linear operators on a complex Hilbert space H. Specifically, for every O∈NA(H) and each ϵ>0, there exists O′∈B(H) such that ‖O−O′‖<ϵ. This result is achieved through the convergence of sequences and the existence of limit points. The properties A and B of Lindenstrauss ensure the density of NA(H), and we show that countable unions, finite intersections, countable tensor products, and countable Cartesian products preserve density in the associated classes. Furthermore, density in NA(H) exhibits transitivity. Building on this foundation of density, we next investigate the concept of dentability in norm-attainable classes within the Banach algebra B(H). Dentability, which is closely related to density via Radon–Nikodým property, refers to the existence of a bounded linear norm-attainable operator within the class that lies outside the closed convex hull of the subclass obtained by excluding a sufficiently small ball around this operator. We provide conditions for dentability and s-dentability in subclasses, closures, closed convex hulls, and superclasses of norm-attainable classes. Moreover, we demonstrate that countable unions, Cartesian products, and finite intersections preserve dentability. We also prove that arbitrary unions, finite intersections, and arbitrary Cartesian products maintain the dentability of classes. Our work significantly contributes to the characterization and understanding of dentability in norm-attainable classes. The findings advance knowledge in operator analysis, operator theory, and optimization, particularly in relation to dentability. These results enhance the study of the lineability and spaceability of norm-attainable classes and Banach spaces.

Keywords

• Norm-attainable class
• Density
• Metric
• Topology
• Limit point
• Dentability