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

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