Some relations between Schwarz-Pick inequality and von Neumann's inequality

We study a Schwarz-Pick type inequality for the Schur-Agler class $SA(B_{\delta})$. In our operator theoretical approach, von Neumann's inequality for a class of generic tuples of $2\times 2$ matrices plays an important role rather than holomorphy. In fact, the class $S_{2, gen}(B_{\Delta})$ consisting of functions that satisfy the inequality for those matrices enjoys \begin{equation*} d_{\mathbb{D}}(f(z), f(w))\le d_{\Delta}(z, w) \;\;(z,w\in B_{\Delta}, f\in S_{2, gen}(B_{\Delta})). \end{equation*} Here, $d_{\Delta}$ is a function defined by a matrix $\Delta$ of abstract functions. Later, we focus on the case when $\Delta$ is a matrix of holomorphic functions. We use the pseudo-distance $d_{\Delta}$ to give a sufficient condition on a diagonalizable commuting tuple $T$ acting on $\mathbb{C}^2$ for $B_{\Delta}$ to be a complete spectral domain for $T$. We apply this sufficient condition to generalizing von Neumann's inequalities studied by Drury and by Hartz-Richter-Shalit.