Frechet differentiability and quasi-polyhedrality in spaces of operators

TL;DR

This paper investigates conditions under which very smooth points in the space of compact operators between certain Banach spaces are also Frechet smooth in the larger space of bounded operators, linking smoothness properties to differentiability.

Contribution

It identifies classes of Banach spaces where smooth points in compact operators imply Frechet smoothness in the space of all bounded operators, advancing understanding of operator space geometry.

Findings

Very smooth points in compact operators are Frechet smooth in the space of bounded operators for certain Banach spaces.

Smoothness in higher duals can lead to Frechet differentiability in operator spaces.

The paper provides conditions connecting smoothness properties of Banach spaces to differentiability of operator norms.

Abstract

Let $X, Y$ be infinite dimensional, Banach spaces. Let $\mathcal{L}(X, Y)$ be the space of bounded operators . Motivated by the fact that smoothness of norm in the higher duals of even order of a Banach space can lead to Frechet differentiability, we exhibit classes of Banach spaces $X, Y$ where very smooth points (i.e., smooth points that remain smooth in the bidual) in the space of compact operators $\mathcal{K}(X, Y)$ are Frechet smooth in $\mathcal{L}(X, Y)$ and hence in $\mathcal{K}(X, Y)$.