Coefficient multipliers of Bloch functions

Anderson, J. M.; Shields, A. L.

doi:10.1090/S0002-9947-1976-0419769-6

Coefficient multipliers of Bloch functions
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by J. M. Anderson and A. L. Shields PDF

Trans. Amer. Math. Soc. 224 (1976), 255-265 Request permission

Abstract:

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

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