Hypercyclicity of adjoint of convex weighted shift and multiplication operators on Hilbert spaces

Document Type : Original Article


Department of Basic Science, Hamedan University of Technology, Hamedan, Iran.


A bounded linear operator $T$ on a Hilbert space $\mathfrak{H}$ is
convex, if
$$\|\mathfrak{T}^{2}v\|^2-2\|\mathfrak{T}v\|^2+\|v\|^2 \geq 0.$$
In this paper, sufficient conditions to hypercyclicity of adjoint of unilateral (bilateral) forward (backward) weighted shift operator is given. Also, we present some example of convex operators such that it's adjoint is hypercyclic. Finally, the spectrum of convex multiplication operators is obtained and an example of convex, multiplication operators is given such that it's adjoint is hypercyclic.


[1] M.F. Ahmadi, K. Hedayatian, Supercyclicity of two-isometries, Honam Mathematical Journal, 30(1)
2008, 115-118.
[2] M.F. Ahmadi, K. Hedayatian, Hypercyclicity and supercyclicity of m-isometric opera- tors, The Rocky
Mountain Journal of Mathematics, 2012, 15-23
[3] S.I. Ansari, P.S. Bourdon, Some properties of cyclic operators, Acta Scientiarum Math- ematicarum,
63(1) 1997, 195-208.
[4] F. Bayart, m-isometries on banach spaces, Mathematische Nachrichten 284 (17-18) 2011, 2141-2147.
[5] F. Bayart, E. Matheron, Dynamics of linear operators, 179. Cambridge university press 2009.
[6] T. Bermdez, A. Bonilla, A. Peris, On hypercyclicity and supercyclicity criteria, Bul- letin of the
Australian Mathematical Society, 70(1) 2004, 45-54.
[7] T. Bermdez, I. Marrero, A. Martinn, On the orbit of an m-isometry, Integral Equa- tions and Operator
Theory, 64(4) 2009, 487-494.
[8] K.C. Chan, I. Seceleanu, Orbital limit points and hypercyclicity of operators on analytic function
spaces, Mathematical Proceedings of the Royal Irish Academy, 2010, 99-109.
[9] N. Feldman, Hypercyclicity and supercyclicity for invertible bilateral weighted shifts, Proceedings of
the American Mathematical Society, 131(2) 2003, 479-485.
[10] E.A. Gallardo-Gutirrez, A. Montes-Rodrguez, The role of the angle in supercyclic behavior, Journal
of Functional Analysis, 203(1) 2003, 27-43.
[11] G. Godefroy, J.H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, Journal of Func-
tional Analysis, 98(2) 1991, 229-269.
[12] K.G. Grosse-Erdmann, A.P. Manguillot, Linear chaos, Springer Science and Business Media, 2011.
[13] K. Hedayatian, L. Karimi, On convexity of composition and multiplication operators on weighted
hardy spaces, Abstract and Applied Analysis, 2009 2009.
[14] K. Hedayatian, L. Karimi, Supercyclicity of convex operators, 2018.
[15] D.A Herrero, Hypercyclic operators and chaos, Journal of Operator Theory, 1992, 93-103.
[16] H. Hilden, L. Wallen, Some cyclic and non-cyclic vectors of certain operators, Indiana University
Mathematics Journal, 23(7) 1974, 557-565.
[17] Z.J. Jabonski, J. Stochel, Unbounded 2-hyperexpansive operators, Proceedings of the Edinburgh
Mathematical Society, 44(3) 2001, 613-629.
[18] L. Karimi , M. Faghih Ahmadi, K. Hedayatian, Some properties of concave operators, Turkish Journal
of Mathematics, 40(6) 2016, 1211-1220.
[19] H.N. Salas, Hypercyclic weighted shifts, Transactions of the American Mathematical Society, 347(3)
1995, 993-1004.
[20] H.N. Salas, Supercyclicity and weighted shifts, Studia Math, 135(1) 1999, 55-74.
[21] A.L. Shields, Weighted shift operators and analytic function theory, 1974.
Volume 2, Issue 4
December 2021
Pages 52-59
  • Receive Date: 19 September 2021
  • Revise Date: 07 November 2021
  • Accept Date: 09 November 2021
  • First Publish Date: 09 November 2021