Multipliers and linear functionals for the class

Author:
Niro Yanagihara

Journal:
Trans. Amer. Math. Soc. **180** (1973), 449-461

MSC:
Primary 30A78

MathSciNet review:
0338382

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Abstract | References | Similar Articles | Additional Information

Abstract: Multipliers for the classes are studied recently by several authors, see Duren's book, *Theory of spaces*, Academic Press, New York, 1970. Here we consider corresponding problems for the class of holomorphic functions in the unit disk such that

Our results are:

1. is an -space in the sense of Banach with the distance function

2. A complex sequence is a multiplier for into for a fixed , if and only if for a positive constant .

3. A continuous linear functional on the space is represented by a holomorphic function which satisfies for a positive constant .

Conversely, such a function defines a continuous linear functional on the space .

**[1]**Nelson Dunford and Jacob T. Schwartz,*Linear Operators. I. General Theory*, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR**0117523****[2]**P. L. Duren,*On the multipliers of 𝐻^{𝑝} spaces*, Proc. Amer. Math. Soc.**22**(1969), 24–27. MR**0241651**, 10.1090/S0002-9939-1969-0241651-X**[3]**-,*Theory of spaces*, Pure and Appl. Math., vol. 38, Academic Press, New York, 1970. MR**42**#3552.**[4]**P. L. Duren, B. W. Romberg, and A. L. Shields,*Linear functionals on 𝐻^{𝑝} spaces with 0<𝑝<1*, J. Reine Angew. Math.**238**(1969), 32–60. MR**0259579****[5]**Theodore W. Gamelin,*Uniform algebras*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR**0410387****[6]**T. Gamelin and G. Lumer,*Theory of abstract Hardy spaces and the universal Hardy class*, Advances in Math.**2**(1968), 118–174. MR**0226392****[7]**I. I. Privalov,*Graničnye svoĭstva analitičeskih funkciĭ*, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian). 2d ed.]. MR**0047765****[8]**N. Yanagihara,*Mean growth and Taylor coefficients of some classes of functions*(to appear).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0338382-X

Keywords:
The class ,
zV as an -space in the sense of Banach,
multiplier as a closed operator,
local unboundedness of the space ,
representations of continuous linear functionals on the space

Article copyright:
© Copyright 1973
American Mathematical Society