Some Relations Between Schwarz–Pick Inequality and von Neumann’s Inequality

We study a Schwarz–Pick type inequality for the Schur–Agler class \(SA(B_{\delta })\) $SA(B_{\delta })$ . In our operator theoretical approach, von Neumann’s inequality for a class of generic tuples of \(2\times 2\) $2\times 2$ matrices plays an important role rather than holomorphy. In fact, the class \(S_{2, gen}(B_{\Delta })\) $S_{2,gen}(B_{\mathrm{\Delta }})$ consisting of functions that satisfy the inequality for those matrices enjoys \(\begin{aligned} d_{\mathbb {D}}(f(z), f(w))\le d_{\Delta }(z, w) \;\;(z,w\in B_{\Delta }, f\in S_{2, gen}(B_{\Delta })). \end{aligned}\) $$\begin{array}{c}\hfill d_{\mathbb{D}}(f(z),f(w))\le d_{\mathrm{\Delta }}(z,w)(z,w\in B_{\mathrm{\Delta }},f\in S_{2,gen}(B_{\mathrm{\Delta }})).\end{array}$$ Here, \(d_{\Delta }\) $d_{\mathrm{\Delta }}$ is a function defined by a matrix \(\Delta \) $\mathrm{\Delta }$ of functions. Later, we focus on the case when \(\Delta \) $\mathrm{\Delta }$ is a matrix of holomorphic functions. We use the pseudo-distance \(d_{\Delta }\) $d_{\mathrm{\Delta }}$ to give a sufficient condition on a diagonalizable commuting tuple T acting on \(\mathbb {C}^2\) ${\mathbb{C}}^2$ for \(B_{\Delta }\) $B_{\mathrm{\Delta }}$ to be a complete spectral domain for T. We apply this sufficient condition to generalizing von Neumann’s inequalities studied by Drury (In: Blei RC, Sidney SJ (eds) Banach spaces, harmonic analysis, and probability theory, lecture notes in mathematics, vol 995. Springer, Berlin, pp 14–32, 1983) and by Hartz–Richter–Shalit (Math Z 301:3877–3894, 2022).