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

Abstract

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

$$\begin{aligned} d_{\mathbb {D}}(f(z), f(w))\le d_{\Delta }(z, w) \;\;(z,w\in B_{\Delta }, f\in S_{2, gen}(B_{\Delta })). \end{aligned}$$

Here, $d_{\Delta }$ is a function defined by a matrix $\Delta$ of 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 (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).