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

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 $x\mapsto (1-\Vert x{\Vert }^2)\mathcal{R}f(x)$ 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 ${\rho }_E(zx,zy)\le |z|{\rho }_E(x,y)$ for $x,y\in B_E$ under some conditions on $z\in \mathbb{C}$. Lipschitz continuity of $x\mapsto (1-\Vert x{\Vert }^2)\mathcal{R}f(x)$ 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.