The little Bloch space, B0, is the space of all holomorphic functions f on the unit disk such that limlzl→1 lf'(z)l (1- lzl2) = 0. Finite Blaschke products are clearly in B0, but examples of infinite products in B0 are more difficult to obtain (there are now several constructions due to Sarason, Stephenson and the author, among others). Stephenson has asked whether B0 contains an infinite, indestructible Blaschke product, i.e., a Blaschke product B so that (B(z) - a)/(1 - âB(z)), is also a Blaschke product for every element a ∈ 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 ll f ll∞ = 1 and whose range set, R(f,a) = {w : there exists zn → a, f(zn) = w}, equals the open unit disk for every a ∈ T.
@article{urn:eudml:doc:41531,
title = {An indestructible Blaschke product in the little Bloch space.},
journal = {Publicacions Matem\`atiques},
volume = {37},
year = {1993},
pages = {95-109},
mrnumber = {MR1240926},
zbl = {0810.30024},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41531}
}
Bishop, Christopher J. An indestructible Blaschke product in the little Bloch space.. Publicacions Matemàtiques, Tome 37 (1993) pp. 95-109. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41531/