Density and Dentability in Norm-Attainable Classes

We establish the norm-denseness of the norm-attainable class $NAB(H)$ in the Banach algebra $B(H)$, which consists of all bounded linear operators on a complex Hilbert space $H$. Specifically, for every $O in NAB(H)$ and each $epsilon>0$, there exists $O' in B(H)$ such that $|O - O'| < epsilon$. We achieve this characterization by utilizing the convergence of sequences and the existence of limit points. The properties $A$ and $B$ of Lindenstrauss are sufficient to ensure the density of $NAB(H)$. Moreover, countable unions, finite intersections, countable tensor products, and countable Cartesian products preserve density in the associated classes $NAB(H)$. Additionally, density in $NAB(H)$ exhibits transitivity. We also investigate the concept of dentability in norm-attainable classes defined on the Banach algebra of all bounded linear operators on a complex Hilbert space $H$. Dentability of a norm-attainable class 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 ball of sufficiently small radius containing the particular bounded linear norm-attainable operator. We provide conditions for dentability and $s$-dentability of subclasses, closures, closed convex hulls, and superclasses of given norm-attainable classes. Furthermore, we demonstrate that countable unions, Cartesian products, and finite intersections preserve dentability. Moreover, we 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 of our study advance knowledge and have practical applications in the fields of operator analysis, operator theory, and optimization with respect to dentability. These results enhance the understanding and further characterization of bounded linear operators. Moreover, the findings are valuable in studying the linearbility and spaceability of norm-attainable classes and Banach spaces.