Abstract
Let X be a Banach space of analytic functions on the open unit disk and Γ a subset of linear isometries on X. Sufficient conditions are given for non-supercyclicity of Γ. In particular, we show that the semigroup of linear isometries on the spaces S p (p > 1), the little Bloch space, and the group of surjective linear isometries on the big Bloch space are not supercyclic. Also, we observe that the groups of all surjective linear isometries on the Hardy space H p or the Bergman space L p a (1 < p < ∞, p ≠ 2) are not supercyclic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. I. Ansari: Hypercyclic and cyclic vectors. J. Funct. Anal. 128 (1995), 374–383.
F. Bayart, É. Matheron: Dynamics of Linear Operators. Cambridge Tracts in Mathematics 179, Cambridge University Press, Cambridge, 2009.
J. Becerra Guerrero, A. Rodríguez-Palacios: Transitivity of the norm on Banach spaces. Extr. Math. 17 (2002), 1–58.
J. Bonet, M. Lindström, E. Wolf: Isometric weighted composition operators on weighted Banach spaces of type H ∞. Proc. Am. Math. Soc. 136 (2008), 4267–4273.
P. S. Bourdon, N. S. Feldman: Somewhere dense orbits are everywhere dense. Indiana Univ. Math. J. 52 (2003), 811–819.
J. A. Conejero, V. Müller, A. Peris: Hypercyclic behaviour of operators in a hypercyclic C 0-semigroup. J. Funct. Anal. 244 (2007), 342–348.
J. B. Conway: Functions of One Complex Variable. Graduate Texts in Mathematics 11, Springer, New York, 1978.
E. T. Copson: Asymptotic Expansions. Cambridge Tracts in Mathematics and Mathematical Physics 55, Cambridge University Press, New York, 1965.
R. J. Fleming, J. E. Jamison: Isometries on Banach Spaces. Vol. 2: Vector-valued Function Spaces. Monographs and Surveys in Pure and Applied Mathematics 138, Chapman and Hall/CRC, Boca Raton, 2007.
R. J. Fleming, J. E. Jamison: Isometries on Banach Spaces. Vol. 1: Function Spaces. Monographs and Surveys in Pure and Applied Mathematics 129, Chapman and Hall/CRC, Boca Raton, 2003.
L.-G. Geng, Z.-H. Zhou, X.-T. Dong: Isometric composition operators on weighted Dirichlet-type spaces. J. Inequal. Appl. (electronic only) 2012 (2012), Article No. 23, 6 pages.
P. Greim, J. E. Jamison, A. Kamińska: Almost transitivity of some function spaces. Math. Proc. Camb. Philos. Soc. 116 (1994), 475–488; corrigendum ibid. 121 (1997), 191.
W. Hornor, J. E. Jamison: Isometries of some Banach spaces of analytic functions. Integral Equations Oper. Theory 41 (2001), 410–425.
K. Jarosz: Any Banach space has an equivalent norm with trivial isometries. Isr. J. Math. 64 (1988), 49–56.
C. Kitai: Invariant Closed Sets for Linear Operators. ProQuest LLC, Ann Arbor, University of Toronto, Toronto, Canada, 1982.
F. León-Saavedra, V. Müller: Rotations of hypercyclic and supercyclic operators. Integral Equations Oper. Theory 50 (2004), 385–391.
M. J. Martín, D. Vukotić: Isometries of some classical function spaces among the composition operators. Recent Advances in Operator-Related Function Theory (A. L. Matheson et al., eds.). Proc. Conf., Dublin, Ireland, 2004, Contemp. Math. 393, American Mathematical Society, Providence, 2006, pp. 133–138.
W. P. Novinger, D. M. Oberlin: Linear isometries of some normed spaces of analytic functions. Can. J. Math. 37 (1985), 62–74.
S. Rolewicz: On orbits of elements. Stud. Math. 32 (1969), 17–22.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was in part supported by a grant from Shiraz University Research Council.
Rights and permissions
About this article
Cite this article
Moradi, A., Hedayatian, K., Robati, B.K. et al. Non supercyclic subsets of linear isometries on Banach spaces of analytic functions. Czech Math J 65, 389–397 (2015). https://doi.org/10.1007/s10587-015-0184-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-015-0184-3