Bloch functions on the unit ball of a Banach space

Abstract

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