Hypercyclic behavior of translation operators on spaces of analytic functions on Hilbert spaces. (English) Zbl 1343.47007

Hypercyclity is one of the main concepts in linear dynamics. An operator \(T:X \to X\) on a Fréchet space is hypercyclic if there exists a vector \(x \in X\) whose orbit \(\{x, Tx, T^{2}x, \ldots \}\) is dense in \(X\). A result by G. D. Birkhoff in [C. R. Acad. Sci., Paris 189, 473–475 (1929; JFM 55.0192.07)] shows that the translation operator \(T_{a} : f(z) \mapsto f(z+a)\) (with \(a \in \mathbb{C}\)) from the space of entire functions \(H(\mathbb{C})\) into itself is hypercyclic. Since then, the hypercyclity of this operator on different spaces has received the attention of several mathematicians.
The present paper goes on this direction, considering the translation operator on a certain Hilbert space: a generalized symmetric Fock space. This is defined from a Hilbert space \(E\) as the Hilbert direct sum (with some Hilbert norm \(\| \cdot \|_{\eta}\)) of the symmetric tensor powers \[ \mathcal{F}_{\eta} = \mathbb{C} \oplus E \oplus \otimes_{s}^{2}E \oplus \otimes_{s}^{3}E \oplus \cdots \] Conditions are given so that the space \(\mathcal{H}_{\eta}= \mathcal{F}_{\eta}^{*}\) consists of entire functions on \(E\).
Then the main result of the paper analyzes when the translation operator \(T_{a} : \mathcal{H}_{\eta} \to \mathcal{H}_{\eta}\) (with \(a \in E\)) is hypercyclic. As a preceding result, a characterization for the differentiation operator \(D_{a}\) to be bounded on \(\mathcal{H}_{\eta}\) is given.


47A16 Cyclic vectors, hypercyclic and chaotic operators
47B33 Linear composition operators
46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces


JFM 55.0192.07
Full Text: DOI


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