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Hardy spaces and Jensen measures


Author: Takahiko Nakazi
Journal: Trans. Amer. Math. Soc. 274 (1982), 375-378
MSC: Primary 46J10
DOI: https://doi.org/10.1090/S0002-9947-1982-0670939-4
MathSciNet review: 670939
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Abstract: Suppose $ A$ is a subalgebra of $ {L^\infty }(m)$ on which $ m$ is multiplicative. In this paper, we show that if $ m$ is a Jensen measure and $ A + \overline A $ is dense in $ {L^2}(m)$, then $ A + \overline A $ is weak-* dense in $ {L^\infty }(m)$, that is, $ A$ is a weak-* Dirichlet algebra. As a consequence of it, it follows that if $ A + \overline A $ is dense in $ {L^4}(m)$, then $ A$ is a weak-* Dirichlet algebra. (It is known that even if $ A + \overline A $ is dense in $ {L^3}(m)$, $ A$ is not a weak-* Dirichlet algebra.) As another consequence, it follows that if $ B$ is a subalgebra of the classical Hardy space $ {H^\infty }$ containing the constants and dense in $ {H^2}$, then $ B$ is weak-* dense in $ {H^\infty }$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0670939-4
Keywords: Hardy spaces, Jensen measures, weak-* Dirichlet algebras, backward shift invariant subalgebras
Article copyright: © Copyright 1982 American Mathematical Society

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