Illinois Journal of Mathematics

Common hypercyclic vectors for certain families of differential operators

N. Tsirivas

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $(k_{n})$ be a strictly increasing sequence of positive integers. If $\sum_{n=1}^{+\infty}\frac{1}{k_{n}}$ $=+\infty$, we establish the existence of an entire function $f$ such that for every $\lambda\in(0,+\infty)$ the set $\{\lambda^{k_{n}}f^{(k_{n})}(\lambda z):n=1,2,\ldots\}$ is dense in the space of entire functions endowed with the topology of uniform convergence on compact subsets of the complex plane. This provides the best possible strengthened version of a corresponding result due to Costakis and Sambarino (Adv. Math. 182 (2004) 278–306). From this, and using a non-trivial result of Weyl which concerns the uniform distribution modulo $1$ of certain sequences, we also derive an entire function $g$ such that for every $\lambda\in J$ the set $\{\lambda^{k_{n}}g^{(k_{n})}(\lambda z):n=1,2,\ldots\}$ is dense in the space of entire functions, where $J$ is “almost” equal to the set of non-zero complex numbers. On the other hand, if $\sum_{n=1}^{+\infty}\frac{1}{k_{n}}<+\infty$ we show that the conclusions in the above results fail to hold.

Article information

Source
Illinois J. Math., Volume 60, Number 3-4 (2016), 625-640.

Dates
Received: 20 June 2015
Revised: 7 December 2016
First available in Project Euclid: 22 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1506067283

Digital Object Identifier
doi:10.1215/ijm/1506067283

Mathematical Reviews number (MathSciNet)
MR3707638

Zentralblatt MATH identifier
06790319

Subjects
Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators

Citation

Tsirivas, N. Common hypercyclic vectors for certain families of differential operators. Illinois J. Math. 60 (2016), no. 3-4, 625--640. doi:10.1215/ijm/1506067283. https://projecteuclid.org/euclid.ijm/1506067283


Export citation

References

  • E. Abakumov and J. Gordon, Common hypercyclic vectors for multiples of backward shift, J. Funct. Anal. 200 (2003), 494–504.
  • F. Bayart, Common hypercyclic vectors for high dimensional families of operators, Int. Math. Res. Not. IMRN 2016 (2016), 6512–6552.
  • F. Bayart and G. Costakis, Hypercyclic operators and rotated orbits with polynomial phases, J. Lond. Math. Soc. (2) 89 (2014), 663–679.
  • F. Bayart and E. Matheron, How to get common universal vectors, Indiana Univ. Math. J. 56 (2007), 553–580.
  • F. Bayart and E. Matheron, Dynamics of linear operators, Cambridge Tracts in Math., vol. 179, Cambridge Univ. Press, Cambridge, 2009.
  • K. C. Chan and R. Sanders, Two criteria for a path of operators to have common hypercyclic vectors, J. Operator Theory 61 (2009), 191–223.
  • K. C. Chan and R. Sanders, An SOT-dense path of chaotic operators with same hypercyclic vectors, J. Operator Theory 66 (2011), 107–124.
  • A. Conejero, V. Müller and A. Peris, Hypercyclic behaviour of operators in a hypercyclic $C_0$-semigroup, J. Funct. Anal. 244 (2007), 342–348.
  • G. Costakis, Approximation by translates of entire functions, Complex and harmonic analysis, Destech Publ., Inc., Lancaster, PA, 2007, pp. 213–219.
  • G. Costakis and M. Sambarino, Genericity of wild holomorphic functions and common hypercyclic vectors, Adv. Math. 182 (2004), 278–306.
  • G. Costakis and N. Tsirivas, Common hypercyclic vectors and universal functions, available at \arxivurlarXiv:1511.06145.
  • G. Costakis, N. Tsirivas and V. Vlachou, Non-existence of common hypercyclic vectors for certain families of translations operators, Comput. Methods Funct. Theory 15 (2015), 393–401.
  • K. Grosse-Erdmann and A. Peris, Linear chaos, Universitext, Springer, London, 2011.
  • L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Dover Publications, Mineola, NY, 2006.
  • F. Leon-Saavedra and V. Müller, Rotations of hypercyclic and supercyclic operators, Integral Equations Operator Theory 50 (2004), 385–391.
  • R. Sanders, Common hypercyclic vectors and the hypercyclicity criterion, Integral Equations Operator Theory 65 (2009), 131–149.
  • S. Shkarin, Universal elements for non-linear operators and their applications, J. Math. Anal. Appl. 348 (2008), 193–210.
  • S. Shkarin, Remarks on common hypercyclic vectors, J. Funct. Anal. 258 (2010), 132–160.
  • N. Tsirivas, Common hypercyclic functions for translation operators with large gaps II, Amer. J. Math. Statist. 6 (2016), 57–70.
  • N. Tsirivas, Common hypercyclic functions for translation operators with large gaps, J. Funct. Anal. 272 (2017), 2726–2751.
  • N. Tsirivas, Common hypercyclic vectors for families of backward shift operators, J. Operator Theory 77 (2017), 3–17.
  • N. Tsirivas, Existence of common hypercyclic vectors for translation operators, available at \arxivurlarXiv:1411.7815.