Subdifferential of the $\mathcal{B}(\mathcal{H},\mathcal{K})$ norm, and approximate orthogonality

Abstract

We present an expression for the right hand derivative of the $\mathcal{B}(\mathcal{H},\mathcal{K})$ norm generalizing the result for $\mathcal{K}=\mathcal{H}$ in [D. J. Ke$\check{\mathrm{c}}$ki$\grave{\mathrm{c}}$, Gateaux derivative of $B(H)$ norm, Proc. Amer. Math. Soc. 133 (2005): 2061--2067]. Using this, we obtain the subdifferential of the $\mathcal{B}(\mathcal{H},\mathcal{K})$ norm. For tuples of operators $\mathbf{A},\mathbf{X}\in$ ${\mathcal{B}(\mathcal{H},\mathcal{H}}^d)$, we give a characterization for $0$ 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, A distance formula for tuples of operators, Linear Algebra Appl. 650 (2022): 267--285]. We define the concept of $\epsilon$-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 $A\in \mathcal{B}(\mathcal{H},\mathcal{K})$ to be $\epsilon$-Birkhoff orthogonal to a subspace of $\mathcal{B}(\mathcal{H},\mathcal{K})$, when $A$ is compact.