An Indestructible Blaschke Product in the Little Bloch Space
نویسنده
چکیده
CHRISTOPHER J . BISHOP The little Bloch space, 130 , is the space of all holomorphic functions f on the unit disk such that lim1 z 1l (f'(z)j(1 Iz12) = 0. Finite Blaschke products are clearly in 130, but examples of infinite products in 80 are more difficult to obtain (there are now several constructions due to Sarason, Stephenson and the author, among others) . Stephenson has asked whether 130 contains an infinite, indestructible Blaschke product, Le., a Blaschke product B so that (B(z) a)/(1 QB(z)), is also a Blaschke product for every a E D. In this paper we give an afirmative answer to his question by constructing such a Blaschke product. We also answer a question of Carmona and Cufí by constructing a VMO function, f, so that Ilf J¡ . = 1 and whose range set, R(f, a) = {w : there exists zn, ~ a, f(z~) = w}, equals the open unit disk for every a E T .
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