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

 

Isometries of the Zygmund $ F$-algebra


Author: Sei-ichiro Ueki
Journal: Proc. Amer. Math. Soc. 140 (2012), 2817-2824
MSC (2010): Primary 32A37; Secondary 47B33
Published electronically: December 28, 2011
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Abstract: In his monograph A. Zygmund introduced the space $ N{\log }^{\alpha }N$
$ (\alpha >0)$ of holomorphic functions on the unit ball that satisfy

$\displaystyle \sup _{0\le r <1} \int _{\mathbb{S}} {\varphi }_{\alpha }(\log (1+\vert f(r\zeta )\vert))d\sigma (\zeta ) < \infty , $

where $ {\varphi }_{\alpha }(t)= t\{\log ({\gamma }_{\alpha }+t)\}^{\alpha }$ for $ t \in [0, \infty )$ and $ {\gamma }_{\alpha }=\max \{e, e^{\alpha }\}$. In 2002, O.M. Eminyan provided some basic properties of $ N{\log }^{\alpha }N$. In this paper we will characterize injective and surjective linear isometries of $ N{\log }^{\alpha }N$. As an application, we will consider isometrically equivalent composition operators or multiplication operators on $ N{\log }^{\alpha }N$.

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Additional Information

Sei-ichiro Ueki
Affiliation: Faculty of Engineering, Ibaraki University, Hitachi 316-8511, Japan
Email: sei-ueki@mx.ibaraki.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11146-8
PII: S 0002-9939(2011)11146-8
Keywords: Isometries, Zygmund $F$-algebra, composition operators
Received by editor(s): October 1, 2010
Received by editor(s) in revised form: March 16, 2011
Published electronically: December 28, 2011
Communicated by: Richard Rochberg
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.