## Osaka Journal of Mathematics

### Hypercyclicity of translation operators in a reproducing kernel Hilbert space of entire functions induced by an analytic Hilbert-space-valued kernel

#### Abstract

The study of the hypercyclicity of an operator is an old problem in mathematics; it goes back to a paper of Birkhoff in 1929 proving the hypercyclicity of the translation operators in the space of all entire functions with the topology of uniform convergence on compact subsets. This article studies the hypercyclicity of translation operators in some general reproducing kernel Hilbert spaces of entire functions. These spaces are obtained by duality in a complex separable Hilbert space $\mathcal{H}$ by means of an analytic $\mathcal{H}$-valued kernel. A link with the theory of de Branges spaces is also established. An illustrative example taken from the Hamburger moment problem theory is included.

#### Article information

Source
Osaka J. Math., Volume 52, Number 3 (2015), 581-601.

Dates
First available in Project Euclid: 17 July 2015

https://projecteuclid.org/euclid.ojm/1437137610

Mathematical Reviews number (MathSciNet)
MR3370467

Zentralblatt MATH identifier
06502586

#### Citation

García, A.G.; Hernández-Medina, M.A.; Portal, A. Hypercyclicity of translation operators in a reproducing kernel Hilbert space of entire functions induced by an analytic Hilbert-space-valued kernel. Osaka J. Math. 52 (2015), no. 3, 581--601. https://projecteuclid.org/euclid.ojm/1437137610

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