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 $1<p,q<\infty$, 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-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.

💡 Research Summary

The paper studies three interrelated properties of bounded linear operators between Banach spaces: the Bhatia‑Šemrl property, strong subdifferentiability of the operator norm, and the condition that the essential norm is strictly smaller than the operator norm. After recalling the Birkhoff‑James orthogonality and its extension to operator spaces, the authors define the Bhatia‑Šemrl property: an operator T satisfies it if for every A with T ⟂_B A there exists a norm‑attaining unit vector x∈M_T such that T x ⟂_B A x. This notion generalizes the original result of Bhatia and Šemrl for Hilbert spaces.

The main contributions are twofold. First, for operators on an infinite‑dimensional Hilbert space H, the authors prove the equivalence \