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

TL;DR

This paper derives the subdifferential of the operator norm on bounded operators between Hilbert spaces, generalizes existing results, and introduces the concept of epsilon-Birkhoff orthogonality in normed spaces.

Contribution

It provides a new expression for the right-hand derivative of the operator norm, characterizes best approximations for operator tuples, and introduces epsilon-Birkhoff orthogonality in general normed spaces.

Findings

Derived the subdifferential of the $\mathcal{B(H,K)}$ norm.

Characterized when zero is a best approximation to operator tuples.

Established conditions for epsilon-Birkhoff orthogonality involving compact operators.

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 $\epsilon$-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 $A\in \mathcal{B(H,K)}$ to be $\epsilon$-Birkhoff orthogonal to a subspace of $\mathcal{B(H,K)}$, when $A$ is compact.