Document Type : Research Paper
Authors
1 Assistant Professor, Department of Mathematics, Farhangian University, Tehran, Iran.
2 Department of Mathematics, Faculty of Science, University of Maragheh, P.O. Box55181-83111, Maragheh, Iran.
Abstract
In this paper, we introduce subspace-frequently hypercyclic operators. We show that these operators are subspace-hypercyclic and there are subspace-hypercyclic operators that are not subspace-frequently hypercyclic. There is a criterion like to subspace-hypercyclicity criterion that implies subspace-frequent hypercyclicity and if an operator $T$ satisfies this criterion, then $T\oplus T$ is subspace-frequently hypercyclic. Additionally, operators on finite spaces can not be subspace-frequently hypercyclic.
Keywords
[2] F. Bayart and S. Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc., 358(11) (2006), pp. 5083-5117.
[3] F. Bayart and S. Grivaux, Invariant Gaussian measures for operators on Banach spaces and linear dynamics, Proc. London Math. Soc., 94(3) (2007), pp. 181-210.
[4] A. Bonilla and K.G. Grosse-Erdmann, Frequently hypercyclic operators and vectors, Ergod. Theor. Dyn. Syst., 27 (2007), pp. 383-404.
[5] S. Grivaux, Frequently hypercyclic operators with irregularly visiting orbits, J. Math. Anal. Appl., 462 (2018), pp. 542-553.
[6] K.G. Grosse-Erdmann, Frequently hypercyclic bilateral shifts, Glasgow. Math. J., 61(2) (2019), pp. 271-286.
[7] K.G. Grosse-Erdmann and A. Peris, Frequently dense orbits, C. R. Acad. Sci. Paris, Ser. I, 341 (2005), pp. 123-128.
[8] K.G. Grosse-Erdmann and A. Peris Manguillot, Linear chaos, Springer, 2011.
[9] B.F. Madore and R.A. Martinez-Avendano, Subspace hypercyclicity, J. Math. Anal. Appl., 373(2) (2011), pp. 502-511.
[10] R.A. Martinez-Avendano and O. Zatarain-Vera, Subspace-hypercyclicity for Toeplitz operators, J. Math. Anal. Appl., 422(1) (2015), pp. 772-775.
[11] Q. Menet, Linear chaos and frequent hypercyclicity, Trans. Amer. Math. Soc., 369(7) (2017), pp. 4977-4994.
[12] H. Rezaei, Notes on subspace-hypercyclic operators, J. Math. Anal. Appl., 397(1) (2013), pp. 428-433.
[13] S. Shkarin, On the spectrum of frequently hypercyclic operators, Proc. Amer. Math. Soc., 137(1) (2009), pp. 123-134.
[14] T.K. Subrahmonian Moothathu, Two remarks on frequent hypercyclicity, J. Math. Anal. Appl., 408 (2013), pp. 843-845.
[15] S. Talebi and M. Moosapoor, Subspace-chaotic operators and subspace-weakly mixing operators, Int. J. of Pure and Applied Math., 78 (2012), pp. 879-885.
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