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 f on the unit ball B E 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 − ‖ x ‖ 2 ) ‖ f ′ ( x ) ‖ (1-\|x\|^2) \|f’(x)\| on B E B_E and by preserving the invariance of the correspondiing seminorm when we compose with automorphisms φ \varphi of B E 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 B_{E} are Bloch functions in both ways.