Super-recurrence for backward shifts
Abstract
In this paper, we characterize the super-recurrence of backward shifts acting on the weighted sequence spaces $\ell^{p}(\mathbb{Z},\nu)$ for $1 \leq p<\infty$ and $c_{0}(\mathbb{Z},\nu),$ where $v:=\left(v_{n}\right)_{n}$is a strictly positive sequence of weights. As a result, we show that supercyclic and super-recurrent backward shifts are equivalent.
We also prove that there are no super-recurrent backward shifts neither on $\ell^{\infty}(\mathbb{N},\nu)$ nor on $\ell^{\infty}(\mathbb{Z},\nu).$
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References
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S. Rolewics, On orbits of elements. Studia Math. 32, 17-22, 1969.
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M. Amouch, O. Benchiheb, Some versions of supercyclicity of a set of operators. Filomat. 35.5, 1619-1627, 2021.
F. Bayart, E. Matheron, Dynamics of linear operators. New York, NY, USA, Cambridge University Press 2009.
A. Bonilla, K. G. Grosse-Erdmann, A. López-Martínez, A. Peris, Frequently recurrent operators. Journal of Functional Analysis, 109713, 2022.
J. Bonet, F. Mart\'{\i}nez-Giménez, A. Peris, Linear chaos on Fréchet spaces. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13.7, 1649-1655, 2003.
G.D. Birkhoff, Surface transformations and their dynamical applications. Acta Math. 43, 1-119, 1920.
G. Costakis, A. Manoussos, I. Parissis, Recurrent linear operators. Complex Anal. Oper. Theory. 8, 1601-1643, 2014.
G. Costakis, I. Parissis, Szemerédi's theorem, frequent hypercyclicity and multiple recurrence. Mathematica Scandinavica. 251--272, 2012.
N. S. Feldman, The dynamics of cohyponormal operators. Contemp. Math. 321, Amer. Math. Soc. 71-85, 2003.
K.-G. Grosse-Erdmann, A. Peris Manguillot, Linear Chaos. Universitext, Springer-Verlag, 2011.
K.-G. Grosse-Erdmann, Universal families and hypercyclic operators. Bull. Amer. Math. Soc. 36, 345-381, 1999.
H.M. Hilden, L.J. Wallen, Some cyclic and non-cyclic vectors of certain operators. Indiana Univ. Math. J. 23, 557-565, 1974.
H. Minkowski, H. Poincaré, Sur le problème des trois corps et les équations de la dynamique. J. F. M. 22, 907-914, 1890.
A. Montes-Rodr\'{\i}guez, H.N. Salas, Supercyclic subspaces, spectral theory and weighted shifts. Adv. Math. 163, 74–134, 2001.
S. Rolewics, On orbits of elements. Studia Math. 32, 17-22, 1969.
H. N. Salas, Supercyclicity and weighted shifts. Studia Math. 135, 55–74, 1999.
Published
2025-12-04
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Research Articles
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