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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Single generator problem


Author: Jun-ichi Tanaka
Journal: Trans. Amer. Math. Soc. 348 (1996), 4113-4129
MSC (1991): Primary 46J10, 46J15, 43A17; Secondary 30H05, 54H20
DOI: https://doi.org/10.1090/S0002-9947-96-01566-8
MathSciNet review: 1340187
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Abstract: Using the Stone-\v{C}ech compactification $\beta \mathbf Z$ of integers, we introduce a free extension of an almost periodic flow. Together with some properties of outer functions, we see that, in a certain class of ergodic Hardy spaces $H^p(\mu )$, $1\le p\le \infty $, the corresponding subspaces $H_0^p(\mu )$ are all singly generated. This shows the existence of maximal weak-$^*$ Dirichlet algebras, different from $H^\infty $ of the disc, for which the single generator problem is settled.


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Additional Information

Jun-ichi Tanaka
Affiliation: Department of Mathematics, Tsuru University, Tsuru City, Yamanashi 402, Japan

DOI: https://doi.org/10.1090/S0002-9947-96-01566-8
Keywords: Ergodic Hardy spaces, outer functions, single generators, cocycles
Received by editor(s): February 3, 1994
Received by editor(s) in revised form: May 24, 1995
Additional Notes: This research was partially supported by Grants 03640174, 04640183 and 05640221 from the Japanese Ministry of Education
Dedicated: Dedicated to Professor Yuji Ito on his 60th birthday
Article copyright: © Copyright 1996 American Mathematical Society