On Bhatia-\v{S}emrl Property, Strong Subdifferentiability and Essential Norm of Operators on Banach Spaces

TL;DR

This paper explores the relationships among the Bhatia-9emrl property, strong subdifferentiability, and essential norm conditions for bounded linear operators on Banach and Hilbert spaces, revealing new characterizations and properties.

Contribution

It establishes new equivalences between essential norm, strong subdifferentiability, and the Bhatia-9emrl property for operators on Hilbert and 9p,9q spaces, and provides examples of such operators.

Findings

Essential norm less than operator norm iff point of strong subdifferentiability with compact norm-attainment set.

Operators with Bhatia-9emrl property have essential norm less than operator norm.

Constructed examples of operators satisfying the Bhatia-9emrl property.

Abstract

We investigate the interplay among three key properties of bounded linear operators between Banach spaces: the Bhatia-\v{S}emrl property, strong subdifferentiability and the condition that the essential norm is strictly less than the operator norm. For a Hilbert space $H$ and for $1<p,q<\infty$, we show that for any operator in $B(H)$ and $B(\ell_p, \ell_q)$, the essential norm is strictly less than the operator norm if and only if it is the point of strong subdifferentiability of the norm and its norm-attainment set is compact. Moreover, for operators in these spaces that satisfy the Bhatia-\v{S}emrl property, we show that their essential norm must be strictly less than their operator norm. We also study norm one projections satisfying the Bhatia-\v{S}emrl property and provide examples of operators that possess this property.