On uniform Bishop-Phelps-Bollob\'as type approximations of linear operators and preservation of geometric properties

TL;DR

This paper investigates how uniform epsilon-BPB approximations of linear operators preserve geometric properties like smoothness and extremality in Banach and Hilbert spaces, improving and generalizing previous results.

Contribution

It introduces new conditions under which geometric properties are preserved during uniform epsilon-BPB approximations, extending prior work and providing concrete examples.

Findings

Many geometric properties are preserved under small epsilon approximations.

Examples of pairs of Banach spaces with non-trivial norm-preserving approximation property.

Generalizations of earlier results on operator approximations and geometric property preservation.

Abstract

We study uniform $\epsilon-$BPB approximations of bounded linear operators between Banach spaces from a geometric perspective. We show that for sufficiently small positive values of $\epsilon,$ many geometric properties like smoothness, norm attainment and extremality of operators are preserved under such approximations. We present examples of pairs of Banach spaces satisfying non-trivial norm preserving uniform $\epsilon-$BPB approximation property in the global sense. We also study these concepts in case of bounded linear operators between Hilbert spaces. Our approach in the present article leads to the improvement and generalization of some earlier results in this context.