Lipschitz continuity of the dilation of Bloch functions on the unit ball of a Hilbert space and applications

TL;DR

This paper proves Lipschitz continuity of a dilation map for Bloch functions on the unit ball of Hilbert spaces, extending classical results and exploring applications in interpolation and composition operators.

Contribution

It extends the Lipschitz continuity result of the dilation map to infinite-dimensional Hilbert spaces and applies it to interpolation and composition operator theory.

Findings

Dilation map is Lipschitz continuous with respect to pseudohyperbolic distance.

Necessary conditions for sequences to be interpolating in (B_E).

Characterization of bounded below composition operators on (B_E).

Abstract

Let $B_E$ be the open unit ball of a complex finite or infinite dimensional Hilbert space. If $f$ belongs to the space $\mathcal{B}(B_E)$ of Bloch functions on $B_E$, we prove that the dilation map given by $x \mapsto (1-\|x\|^2) \mathcal{R} f(x)$ for $x \in B_E$, where $\mathcal{R} f$ denotes the radial derivative of $f$, is Lipschitz continuous with respect to the pseudohyperbolic distance $\rho_E$ in $B_E$, which extends to the finite and infinite dimensional setting the result given for the classical Bloch space $\mathcal{B}$. In order to provide this result, we will need to prove that $\rho_E(zx,zy) \leq |z| \rho_E(x,y)$ for $x,y \in B_E$ under some conditions on $z \in \mathbb{C}$. Lipschitz continuity of $x \mapsto (1-\|x\|^2) \mathcal{R} f(x)$ will yield some applications which also extends classical results from $\mathcal{B}$ to $\mathcal{B}(B_E)$. On the one hand, we supply results on interpolating sequences for $\mathcal{B}(B_E)$: we show that it is necessary for a sequence in $B_E$ to be separated in order to be interpolating for $\mathcal{B}(B_E)$ and we also prove that any interpolating sequence for $\mathcal{B}(B_E)$ can be slightly perturbed and it remains interpolating. On the other hand, after a deep study of the automorphisms of $B_E$, we provide necessary and suficient conditions for a composition operator on $\mathcal{B}(B_E)$ to be bounded below.