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 \(\mathcal {B}\). 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 on interpolating sequences for \(\mathcal {B}(B_E)\) which also extends classical results from \(\mathcal {B}\) to \(\mathcal {B}(B_E)\). Indeed, we show that it is necessary for a sequence in \(B_E\) to be separated 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.