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

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The Bloch space has been studied on the open unit disk of C and some ho- mogeneous domains of C n . We dene Bloch functions on the open unit ball of a Hilbert space E and prove that the corresponding space 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 BE are also Bloch functions. ones, resulting the fact that if for a given n; the restrictions of the function to the n-dimensional subspaces have their Bloch norms uniformly bounded, then the function is a Bloch one and conversely. We also introduce an equivalent norm forB(BE) obtained by replacing the gradient by the radial derivative. We exhibit in Section 3 another equivalent norm forB(BE) which is invariant- modulo the constant functions - under the action of the automorphisms of the ball. This is achieved without appealing to the invariant Laplacian and relying only on properties of automorphisms of BE: Further, we are able to show that the space H 1 (BE) of bounded analytic functions is contractively embedded in B(BE), as it occurs in the nite dimensional case. Examples of unbounded Bloch functions are also shown.