Azimi, M. (2017). Subspace-diskcyclic sequences of linear operators. Sahand Communications in Mathematical Analysis, 8(1), 97-106. doi: 10.22130/scma.2017.23850

Mohammad Reza Azimi. "Subspace-diskcyclic sequences of linear operators". Sahand Communications in Mathematical Analysis, 8, 1, 2017, 97-106. doi: 10.22130/scma.2017.23850

Azimi, M. (2017). 'Subspace-diskcyclic sequences of linear operators', Sahand Communications in Mathematical Analysis, 8(1), pp. 97-106. doi: 10.22130/scma.2017.23850

Azimi, M. Subspace-diskcyclic sequences of linear operators. Sahand Communications in Mathematical Analysis, 2017; 8(1): 97-106. doi: 10.22130/scma.2017.23850

^{}Department of Mathematics, Faculty of Sciences, University of Maragheh, Maragheh, Iran.

Abstract

A sequence $\{T_n\}_{n=1}^{\infty}$ of bounded linear operators on a separable infinite dimensional Hilbert space $\mathcal{H}$ is called subspace-diskcyclic with respect to the closed subspace $M\subseteq \mathcal{H},$ if there exists a vector $x\in \mathcal{H}$ such that the disk-scaled orbit $\{\alpha T_n x: n\in \mathbb{N}, \alpha \in\mathbb{C}, | \alpha | \leq 1\}\cap M$ is dense in $M$. The goal of this paper is the studying of subspace diskcyclic sequence of operators like as the well known results in a single operator case. In the first section of this paper, we study some conditions that imply the diskcyclicity of $\{T_n\}_{n=1}^{\infty}$. In the second section, we survey some conditions and subspace-diskcyclicity criterion (analogue the results obtained by some authors in \cite{MR1111569, MR2261697, MR2720700}) which are sufficient for the sequence $\{T_n\}_{n=1}^{\infty}$ to be subspace-diskcyclic(subspace-hypercyclic).

[1] N. Bamerni, V. Kadets, and A. Kιlιçman, On subspaces diskcyclicity, arXiv:1402.4682 [math.FA], 1-11.

[2] N. Bamerni, V. Kadets, A. Kιlιçman, and M.S.M. Noorani, A review of some works in the theory of diskcyclic operators, Bull. Malays. Math. Sci. Soc., Vol. 39 (2016) 723-739.

[3] F. Bayart and ´E. Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, Vol. 179, Cambridge University Press, Cambridge, 2009.

[4] L. Bernal-Gonz´alez and K.-G. Grosse-Erdmann, The hypercyclicity criterion for sequences of operators, Studia Math., Vol. 157 No. 1 (2003) 17-32.

[5] P.S. Bourdon, Invariant manifolds of hypercyclic vectors, Proc. Amer. Math. Soc., Vol. 118 No. 3 (1993) 845-847.

[6] G. Godefroy and J.H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal., Vol. 98 No. 2 (1991) 229-269.

[8] R.R. Jiménez-Munguía, R.A. Martínez-Avendaño, and A. Peris, Some questions about subspace-hypercyclic operators, J. Math. Anal. Appl., Vol. 408 No. 1 (2013) 209-212.

[9] C. Kitai, Invariant closed sets for linear operators, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)University of Toronto, Canada 1982.

[10] F. León-Saavedra and V. Müller, Hypercyclic sequences of operators, Studia Math., Vol. 175 No.1 (2006) 1-18.

[11] B.F. Madore and R.A. Martínez-Avendaño, Subspace hypercyclicity, J. Math. Anal. Appl., Vol. 373 No.2 (2011) 502-511.

[12] H. Petersson, A hypercyclicity criterion with applications, J. Math. Anal. Appl., Vol. 327 No. 2 (2007) 1431-1443.

[13] H. Rezaei, Notes on subspace-hypercyclic operators, J. Math. Anal. Appl., Vol. 397 No. 1 (2013) 428-433.

[14] Z.J. Zeana, Cyclic Phenomena of operators on Hilbert space, Thesis, University of Baghdad, 2002.