Subdifferential of the $\mathcal{B(H,K)}$ norm, and approximate orthogonality
Priyanka Grover, Krishna Kumar Gupta, Susmita Seal
TL;DR
The work provides a complete description of the right-hand derivative and subdifferential of the operator norm on B(H,K), extending known results from B(H) to operator tuples. It uses spectral constructions (H_δ(A), Λ(A)) and Banach limits to characterize ∂||A|| for all A, and applies these tools to best-approximation problems for tuples and to ε-Birkhoff orthogonality in operator spaces. A key contribution is the explicit subdifferential structure for compact operators and the connections between norm derivatives, joint numerical ranges W_0, and distance to subspaces. The results have implications for optimization and approximation in spaces of bounded operators, including practical criteria for approximate orthogonality in B(H,K).
Abstract
We present an expression for the right hand derivative of the $\mathcal{B(H,K)}$ norm generalizing the result for $\mathbf{K}=\mathbf{H}$ in [D. J. Ke$\check{\mathrm{c}}$ki$\grave{\mathrm{c}}$, {\it Gateaux derivative of $B(H)$ norm}, Proc. Amer. Math. Soc. {\bf 133} (2005): 2061--2067]. Using this, we obtain the subdifferential of the $\mathcal{B(H, K)}$ norm. For tuples of operators $\mathbf{A},\mathbf{X}\in$ $\mathcal{B(H, H}^d)$, we give a characterization for $\boldsymbol 0$ to be a best approximation to the subspace $\mathbb C^d \mathbf{X}$, generalizing a similar result for $\mathbb C^d \mathbf{I}$ in [P. Grover, S. Singla, {\it A distance formula for tuples of operators}, Linear Algebra Appl. {\bf 650} (2022): 267--285]. We define the concept of $ε$-Birkhoff orthogonality to a subspace in a general normed space and derive a characterization in terms of the subdifferential set. Using this, we deduce interesting results for $A\in \mathcal{B(H,K)}$ to be $ε$-Birkhoff orthogonal to a subspace of $\mathcal{B(H,K)}$, when $A$ is compact.