On Bhatia-Šemrl Property, Strong Subdifferentiability and Essential Norm of Operators on Banach Spaces

We investigate the interplay among three key properties of bounded linear operators between Banach spaces: the Bhatia-Šemrl property, strong subdifferentiability and the condition that the essential norm is strictly less than the operator norm. For a Hilbert space $H$ and for $1<p,q<\mathrm{\infty }$, we show that for any operator in $B(H)$ and $B({\mathrm{\ell }}_p,{\mathrm{\ell }}_q)$, 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-Šemrl property, we show that their essential norm must be strictly less than their operator norm. We also study norm one projections satisfying the Bhatia-Šemrl property and provide examples of operators that possess this property.