# N-SUPERCYCLICITY OF AN A-m-ISOMETRY

• HEDAYATIAN, KARIM (Department of Mathematics, College of Sciences, Shiraz University)
• Accepted : 2015.05.06
• Published : 2015.09.25
• 72 4

#### Abstract

An A-m-isometric operator is a bounded linear operator T on a Hilbert space $\mathcal{H}$ satisfying $\sum\limits_{k=0}^{m}(-1)^{m-k}T^{*^k}AT^k=0$, where A is a positive operator. We give sufficient conditions under which A-m-isometries are not N-supercyclic, for every $N{\geq}1$; that is, there is not a subspace E of dimension N such that its orbit under T is dense in $\mathcal{H}$.

#### Keywords

Hilbert space;A-m-isometry;N-supercyclicity

#### Acknowledgement

Supported by : Shiraz University

#### References

1. F. Bayart, E. Matheron, Hyponormal operators, weighted shifts and weak forms of supercyclicity, Proc. Edinb. Math. Soc., (2) 49 (2006), 1-15. https://doi.org/10.1017/S0013091504000975
2. T. Bermudez, I. Marrero, Martinon A., On the orbit of an m-isometry, Integral Equat. Ope. Theorey, 64 (2009), 487-494. https://doi.org/10.1007/s00020-009-1700-3
3. P. S. Bourdon, Orbits of hyponormal operators, Michigan Math. J., 44 (1997), no. 2, 345-353. https://doi.org/10.1307/mmj/1029005709
4. P. S. Bourdon, N. S. Feldman, and J. H. Shapiro, Some properties of N-supercyclic operators, Studia Math. 165 (2004), 135-157. https://doi.org/10.4064/sm165-2-4
5. M. Faghih-Amadi, Powers of A-m-isometric operators and their supercyclicity, to appear in Bull. Malays. Math. Sci. Soc.
6. M. Faghih-Ahmadi, K. Hedayatian, Hypercyclicity and supercyclicity of m-isometric operators, Rocky Mountain J. Math., 42 (2012), 15-23. https://doi.org/10.1216/RMJ-2012-42-1-15
7. N. S. Feldman, N-supercyclic operators, Studia Math., 151 (2002), 141-159. https://doi.org/10.4064/sm151-2-3
8. H. M. Hilden, L. J. Wallen, Some cyclic and non-cyclic vectors of certain operators, Indiana Univ, Math. J., 23 (1973/74), 557-565. https://doi.org/10.1512/iumj.1974.23.23046
9. R. Rabaoui, A. Saddi, On the orbit of an A-m-isometry, Annales Mathematicas Silesianae, 26 (2012), 75-91.
10. O. A. M. Sid Ahmed, A. Saddi, A-m-Isometric operators in semi-Hilbertian spaces, Linear Algebra Appl., 436 (2012), 3930-3942. https://doi.org/10.1016/j.laa.2010.09.012
11. J. Agler, M. Stankus, m-isometric transformations of Hilbert space I, Integral Equat. Ope. Theory 21 (1995), 383-429. https://doi.org/10.1007/BF01222016
12. J. Agler, M. Stankus, m-isometric transformations of Hilbert space II, Integral Equat. Ope. Theory 23 (1995), 1-48. https://doi.org/10.1007/BF01261201
13. J. Agler, M. Stankus, m-isometric transformations of Hilbert space III, Integral Equat. Ope. Theory 24 (1996), 379-421. https://doi.org/10.1007/BF01191619
14. F. Bayart, m-isometries on Banach spaces, Math. Nachr., 284 (2001), 2141-2147.

#### Cited by

1. ( A , m )-isometries on Hilbert spaces vol.540, 2018, https://doi.org/10.1016/j.laa.2017.11.005