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{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, $d_{\Delta }$ 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.

13 pages

References in corpus (2)

• von Neumann's inequality for row contractive matrix tuples
• A polynomial approximation result for free Herglotz-Agler functions

Cited by in corpus (1)

• Three radii associated to Schur functions on the polydisk