Abstract
In this paper, we consider the (weighted) backward shift operators on the double sequence spaces. We establish necessary and sufficient conditions for the pair of backward shifts with constant weights as well as variable weights to be hypercyclic, weakly mixing and mixing.
Similar content being viewed by others
References
Abels, H., Manoussos, A.: Topological generators of abelian Lie groups and hypercyclic finitely generated abelian semigroups of matrices. Adv. Math. 229, 1862–1872 (2012)
Başar, F.: Summability theory and its applications. Bentham Science Publishers, e-books, Monograph, Istanbul-2012, ISBN: 978-1-60805-420-6
Başarır, M., Solancan, O.: On some double sequence spaces. Indian Acad. Math. 21, 193–200 (1999)
Başar, F., Sever, Y.: The space \({\cal{L}}_q\) of double sequences. Math. J. Okayama Univ. 51, 149–157 (2009)
Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)
Bernal-Gonzlez, L.: Hypercyclic sequences of differential and antidifferential operators. J. Approx. Theory 96, 323–337 (1999)
Bourdon, P.S., Shapiro, J.H.: Hypercyclic operators that commute with the Bergman backward shift. Trans. Am. Math. Soc. 352, 5293–5316 (2000)
Bromwich, T.J.I.A.: An Introduction to the Theory of Infinite Series. MacMillan, London (1959)
Costakis, G., Hadjiloucas, D., Manoussos, A.: Dynamics of tuples of matrices. Proc. Am. Math. Soc. 137, 1025–1034 (2009)
Costakis, G., Parissis, I.: Dynamics of tuples of matrices in Jordan form. Oper. Matrices 7, 131–157 (2013)
Delaubenfels, R., Emamirad, H.: Chaos for functions of discrete and continuous weighted shift operators. Ergod. Theory Dynam. Syst. 21, 1411–1427 (2001)
Devaney, R.: A First Course in Chaotic dynamical Systems. Addison-Wesley, Reading (1992)
Feldman, N.S.: Hypercyclic tuples of operators and somewhere dense orbits. J. Math. Anal. Appl. 346, 82–98 (2008)
Gethner, R.M., Shapiro, J.H.: Universal vectors for operators on spaces of holomorphic functions. Proc. Am. Math. Soc. 100, 281–288 (1987)
Godefroy, G., Shapiro, J.H.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98, 229–269 (1991)
Gökhan, A., Çolak, R.: Double sequence space \(l^{\infty }_{2}(p)\). Appl. Math. Comput. 160, 147–153 (2005)
Grosse-Erdmann, K.G., Peris, A.: Linear Chaos. Springer, London (2011)
Hardy, G.H.: On the convergence of certain multiple series. Proc. Camb. Phil. Soc. 2, 124–128 (1904)
Kitai, C.: Invariant closed sets for linear operators. Ph. D. thesis, University of Toronto (1984)
Móricz, F.: Extensions of the spaces \(c\) and \(c_{0}\) from single to double sequences. Acta Math. Hung. 57, 129–136 (1991)
Móricz, F., Rhoades, B.E.: Almost convergence of double sequences and strong regularity of summability matrices. Math. Proc. Cambridge Philos. Soc. 104, 283–294 (1988)
Mursaleen, Edely, Osama H.H.: Statistical convergence of double sequences. J. Math. Anal. Appl. 288, 223–231 (2003)
Rolewicz, S.: On orbits of elements. Stud. Math. 32, 17–22 (1969)
Salas, H.N.: Hypercyclic weighted shifts. Trans. Am. Math. Soc. 347, 993–1004 (1995)
Shkarin, S.: Hypercyclic tuples of operators on \({\mathbb{C}}^n\) and \({\mathbb{R}}^n\). Linear Multilinear Algebra 60, 885–896 (2012)
Yesilkayagil, M., Basar, F.: Domain of Riesz mean in the space \({\cal{L}}_s\). Filomat (2016), to appear
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shu, Y., Wang, W. & Zhao, X. Backward Shifts on Double Sequence Spaces. Results Math 72, 793–811 (2017). https://doi.org/10.1007/s00025-017-0658-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-017-0658-8