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

 

Multipliers and linear functionals for the class $ N\sp{+}$


Author: Niro Yanagihara
Journal: Trans. Amer. Math. Soc. 180 (1973), 449-461
MSC: Primary 30A78
MathSciNet review: 0338382
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Abstract: Multipliers for the classes $ {H^p}$ are studied recently by several authors, see Duren's book, Theory of $ {H^p}$ spaces, Academic Press, New York, 1970. Here we consider corresponding problems for the class $ {N^ + }$ of holomorphic functions in the unit disk such that

$\displaystyle \mathop {\lim }\limits_{r \to 1} \int_0^{2\pi } {{{\log }^ + }} \... ...= \int_0^{2\pi } {{{\log }^ + }\vert f({e^{i\theta }})\vert} d\theta < \infty .$

Our results are:

1. $ {N^ + }$ is an $ F$-space in the sense of Banach with the distance function

$\displaystyle \rho (f,g) = \frac{1}{{2\pi }}\int_0^{2\pi } {\log (1 + \vert f({e^{i\theta }}) - g({e^{i\theta }})\vert)} d\theta .$

2. A complex sequence $ \Lambda = \{ {\lambda _n}\} $ is a multiplier for $ {N^ + }$ into $ {H^q}$ for a fixed $ q,0 < q < \infty $, if and only if $ {\lambda _n} = O(\exp [ - c\sqrt n ])$ for a positive constant $ c$.

3. A continuous linear functional $ \phi $ on the space $ {N^ + }$ is represented by a holomorphic function $ g(z) = \Sigma {b_n}{z^n}$ which satisfies $ {b_n} = O(\exp [ - c\sqrt n ])$ for a positive constant $ c$.

Conversely, such a function $ g(z) = \Sigma {b_n}{z^n}$ defines a continuous linear functional on the space $ {N^ + }$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0338382-X
PII: S 0002-9947(1973)0338382-X
Keywords: The class $ {N^ + },{N^ + }$, zV as an $ F$-space in the sense of Banach, multiplier as a closed operator, local unboundedness of the space $ {N^ + }$, representations of continuous linear functionals on the space $ {N^ + }$
Article copyright: © Copyright 1973 American Mathematical Society