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

We present an expression for the right hand derivative of the $\mathcal{B(H,K)}$ 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 $\mathcal{B(H, K)}$ 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, 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 $A\in \mathcal{B(H,K)}$ to be $\epsilon$-Birkhoff orthogonal to a subspace of $\mathcal{B(H,K)}$, when $A$ is compact.