Coefficient multipliers of Bloch functions

Abstract

The class B \mathcal {B} of Bloch functions is the class of all those analytic functions in the open unit disc for which the maximum modulus is bounded by c / ( 1 − r ) c/(1 - r) on | z | ⩽ r |z| \leqslant r . We study the absolute values of the Taylor coefficients of such functions. In particular, we find all coefficient multipliers from l p {l^p} into B \mathcal {B} and from B \mathcal {B} into l p {l^p} . We find the second Köthe dual of B \mathcal {B} and show its relevance to the multiplier problem. We identify all power series ∑ a n z n \sum {a_n}{z^n} such that ∑ w n a n z n \sum {w_n}{a_n}{z^n} is a Bloch function for every choice of the bounded sequence { w n } \{ {w_n}\} . Analogous problems for H p {H^p} spaces are discussed briefly.