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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
DOI: https://doi.org/10.1090/S0002-9939-2011-11146-8
Published electronically: December 28, 2011
MathSciNet review: 2910768
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Abstract | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)

  • 1. R.K. Campbell-Wright, Equivalent composition operators, Integr. Equ. Oper. Theory, 14 (1991), 775-786. MR 1127536 (92h:47037)
  • 2. J. Cima and W.R. Wogen, On isometries of the Bloch spaces, Illinois J. Math., 24 (1980), 313-316. MR 575069 (82m:30052)
  • 3. C.C. Cowen and B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995. MR 1397026 (97i:47056)
  • 4. O.M. Eminyan, Zygmund $ F$-algebras of holomorphic functions in the ball and in the polydisk, Doklady Math., 65 (2002), 353-355. MR 1930097 (2003i:32013)
  • 5. R.J. Fleming and J.E. Jamison, Isometries on Banach Spaces: Function Spaces, Chapman & Hall/CRC, 2003. MR 1957004 (2004j:46030)
  • 6. F. Forelli, The isometries of $ H^p$, Canad. J. Math., 16 (1964), 721-728. MR 0169081 (29:6336)
  • 7. F. Forelli, A theorem on isometries and the application of it to the isometries of $ H^p(S)$ for $ 2<p<\infty $, Canad. J. Math., 25 (1973), 284-289. MR 0318897 (47:7443)
  • 8. N.J. Gal, Isometric equivalence of differentiated composition operators between spaces of analytic functions, Houston J. Math., 33 (2007), 921-926. MR 2335743 (2008e:47087)
  • 9. N.J.  Gal, J.E. Jamison and A.G. Siskakis, Isometric equivalence of integration operators, Complex Anal. Oper. Theory, 4 (2010), 245-255. MR 2652524
  • 10. W. Hornor and J.E. Jamison, Isometrically equivalent composition operators, Contemp. Math. 213, Amer. Math. Soc., 1998, 65-72. MR 1601072 (99b:47041)
  • 11. W. Hornor and J.E. Jamison, Isometries of some Banach spaces of analytic functions, Integr. Equ. Oper. Theory, 41 (2001), 410-425. MR 1857800 (2002h:46035)
  • 12. Y. Iida and N. Mochizuki, Isometries of some $ F$-algebras of holomorphic functions, Arch. Math., 71 (1998), 297-300. MR 1640082 (99h:30050)
  • 13. C.J. Kolaski, Isometries of Bergman spaces over bounded Runge domains, Canad. J. Math., 33 (1981), 1157-1164. MR 638372 (83b:32028)
  • 14. C.J. Kolaski, Isometries of weighted Bergman spaces, Canad. J. Math., 34 (1982), 910-915. MR 672684 (84a:46054)
  • 15. C.J. Kolaski, Surjective isometries of weighted Bergman spaces, Proc. Amer. Math. Soc., 105 (1989), 652-657. MR 953008 (89m:46042)
  • 16. D. deLeeuw, W. Rudin and J. Wermer, The isometries of some function spaces, Proc. Amer. Math. Soc., 11 (1960), 694-698. MR 0121646 (22:12380)
  • 17. W. Rudin, $ L^p$-isometries and equimeasurability, Indiana Univ. Math. J., 25 (1976), 215-228. MR 0410355 (53:14105)
  • 18. W. Rudin, Function Theory on the Unit Ball of $ \mathbb{C}^n$, Springer-Verlag, 1980. MR 601594 (82i:32002)
  • 19. K. Stephenson, Isometries of the Nevanlinna class, Indiana Univ. Math. J., 26 (1977), 307-324. MR 0432905 (55:5885)
  • 20. A.V. Subbotin, Functional properties of Privalov spaces of holomorphic functions in several variables, Mathematical Notes, 65 (1999), 230-237. MR 1706561 (2000g:32005)
  • 21. A.V. Subbotin, Linear isometry groups of Privalov's spaces of holomorphic functions of several variables, Doklady Math., 60 (1999), 77-79. MR 1727309
  • 22. A. Zygmund, Trigonometric series, vol. 2, Cambridge University Press, 1959. MR 0107776 (21:6498)

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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: https://doi.org/10.1090/S0002-9939-2011-11146-8
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.

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