Connections between Essential Norm, Bhatia-Šemrl Property and Strong Subdifferentiability of Operators on Banach Spaces
We study Birkhoff-James orthogonality in spaces of bounded linear operators between Banach spaces through the Bhatia-Šemrl property. We investigate the interplay among three key properties of bounded linear operators: strong subdifferentiability, the Bhatia-Šemrl property, and the condition that the essential norm is strictly less than the operator norm. If $X$ is a Banach space with a $1$-unconditional basis and $Y$ is any Banach space, we show that every nonzero operator $T \in B(X, Y)$ satisfying the Bhatia-Šemrl property satisfies $\|T\|_e<\|T\|$. We identify a class of pairs of Banach spaces $X$ and $Y$ for which the inequality $\|T\|_e<\|T\|$ can be characterized in terms of strong subdifferentiability of the operator norm at $T$ and compactness of the norm attainment set of $T$. We also study norm one projections satisfying the Bhatia-Šemrl property and provide examples of operators that possess this property.