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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The solution sets of extremal problems in $ H\sp 1$


Author: Eric Hayashi
Journal: Proc. Amer. Math. Soc. 93 (1985), 690-696
MSC: Primary 30D55; Secondary 46E99
MathSciNet review: 776204
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Abstract: Let $ u$ be an essentially bounded function on the unit circle $ T$. Let $ {S_u}$ denote the subset of the unit sphere of $ {H^1}$ on which the functional $ F \mapsto \smallint _0^{2\pi }\bar u({e^{it}})F({e^{it}})dt/2\pi $ attains its norm. A complete description of $ {S_u}$ is given in terms of an inner function $ {b_0}$ and an outer fun tion $ {g_0}$ in $ {H^2}$ for which $ g_0^2$ is an exposed point in the unit ball of $ {H^1}$. An explicit description is given for the kernel of an arbitrary Toeplitz operator on $ {H^2}$. The exposed points in $ {H^1}$ are characterized; an example is given of a strong outer function in $ {H^1}$ which is not exposed.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0776204-7
PII: S 0002-9939(1985)0776204-7
Article copyright: © Copyright 1985 American Mathematical Society