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

Abstract

Let BE be the open unit ball of a complex finite- or infinite-dimensional Hilbert space. If f belongs to the space B(BE ) of Bloch functions on BE , we prove that the dilation map given by x → (1 − x 2)R f (x) for x ∈ BE , where R f denotes the radial derivative of f , is Lipschitz continuous with respect to the pseudohyperbolic distance ρE in BE , which extends to the finite- and infinite-dimensional setting the result given for the classical Bloch space B. To provide this result, we will need to prove that ρE (zx,zy) ≤ |z|ρE (x, y) for x, y ∈ BE under some conditions on z ∈ C. Lipschitz continuity of x → (1 − x 2)R f (x) will yield some applications on interpolating sequences for B(BE ) which also extends classical results from B to B(BE ). Indeed, we show that it is necessary for a sequence in BE to be separated to be interpolating for B(BE ) and we also prove that any interpolating sequence for B(BE ) can be slightly perturbed and it remains interpolating.