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 $S_{2,gen}(B_{\mathrm{\Delta }})$ consisting of functions that satisfy the inequality for those matrices enjoys \begin{equation*} d_{$\mathbb{D}$}(f(z), f(w))\le d_Δ(z, w) ;;(z,w\in B_Δ, f\in S_{2, gen}(B_Δ)). \end{equation*} Here, $d_{\mathrm{\Delta }}$ is a function defined by a matrix $\mathrm{\Delta }$ of abstract functions. Later, we focus on the case when $\mathrm{\Delta }$ is a matrix of holomorphic functions. We use the pseudo-distance $d_{\mathrm{\Delta }}$ to give a sufficient condition on a diagonalizable commuting tuple $T$ acting on ${\mathbb{C}}^2$ for $B_{\mathrm{\Delta }}$ 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.