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arXiv · 2512.15208

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

C. R. Jayanarayanan·Rishit R Rajpopat
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Abstract

We study Birkhoff-James orthogonality in spaces of bounded linear operators between Banach spaces through the Bhatia-\v{S}emrl property. We investigate the interplay among three key properties of bounded linear operators: strong subdifferentiability, the Bhatia-\v{S}emrl property, 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 nonzero 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 a point of strong subdifferentiability of the norm and its norm-attainment set is compact. Moreover, for nonzero 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.

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C. R. Jayanarayanan, Rishit R Rajpopat. 2025-12-17. Connections between Essential Norm, Bhatia-\v{S}emrl Property and Strong Subdifferentiability of Operators on Banach Spaces. https://arxiv.org/abs/2512.15208

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