BLOCH FUNCTIONS ON THE UNIT BALL OF A BANACH SPACE

The space of Bloch functions on bounded symmetric domains is extended by considering Bloch functions f on the unit ball B-E of finite and infinite dimensional complex Banach spaces in two different ways: by extending the classical Bloch space considering the boundness of (1 - parallel to x parallel to(2))parallel to f'(x)parallel to on B-E and by preserving the invariance of the correspondiing seminorm when we compose with automorphisms phi of B-E. We study the connection between these spaces proving that they are different in general and prove that all bounded analytic functions on B-E are Bloch functions in both ways.