Bloch functions on the unit ball of an infinite dimensional Hilbert space

The Bloch space has been studied on the open unit disk of CC and some homogeneous domains of CnCn. We define Bloch functions on the open unit ball of a Hilbert space E   and prove that the corresponding space B(BE)B(BE) is invariant under composition with the automorphisms of the ball, leading to a norm that — modulo the constant functions — is automorphism invariant as well. All bounded analytic functions on BEBE are also Bloch functions.