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

\Let BE be the open unit ball of a complex finite- or infinite-dimensional Hilbert space. If f belongs to the space B(B-E) of Bloch functions on B-E, we prove that the dilation map given by x bar arrow (1 - ||x||(2))R f (x) for x is an element of B-E, where Rf 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 B. To provide this result, we will need to prove that rho(E) (zx, zy) <= |z|rho(E) (x, y) for x, y is an element of B-E under some conditions on z is an element of C. Lipschitz continuity of x bar arrow (1 - ||x||(2))R f (x) will yield some applications on interpolating sequences for B(B-E) which also extends classical results from B to B(B-E). Indeed, we show that it is necessary for a sequence in BE to be separated to be interpolating for B(B-E) and we also prove that any interpolating sequence for B(B-E) can be slightly perturbed and it remains interpolating.