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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Coefficient multipliers of Bloch functions


Authors: J. M. Anderson and A. L. Shields
Journal: Trans. Amer. Math. Soc. 224 (1976), 255-265
MSC: Primary 30A78
MathSciNet review: 0419769
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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 $ \vert z\vert \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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DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0419769-6
PII: S 0002-9947(1976)0419769-6
Article copyright: © Copyright 1976 American Mathematical Society