Isometries of the Zygmund $F$-algebra
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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 \[ \sup _{0\le r <1} \int _{\mathbb {S}} {\varphi }_{\alpha }(\log (1+|f(r\zeta )|))d\sigma (\zeta ) < \infty , \] 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 $N{\log }^{\alpha }N$.References
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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
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 2817-2824
- MSC (2010): Primary 32A37; Secondary 47B33
- DOI: https://doi.org/10.1090/S0002-9939-2011-11146-8
- MathSciNet review: 2910768