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**A hypercyclicity criterion for non-metrizable topological vector spaces.**
*(English)*
Zbl 1475.47010

Summary: We provide a sufficient condition for an operator \(T\) on a non-metrizable and sequentially separable topological vector space \(X\) to be sequentially hypercyclic. This condition is applied to some particular examples, namely, a composition operator on the space of real analytic functions on \(\,]0,1[\,\), which solves two problems of J. Bonet and P. Domański [Math. Proc. Camb. Philos. Soc. 153, No. 3, 489–503 (2012; Zbl 1272.47037)], and the “snake shift” constructed in [J. Bonet et al., Bull. Lond. Math. Soc. 37, No. 2, 254–264 (2005; Zbl 1150.47005)] on direct sums of sequence spaces. The two examples have in common that they do not admit a densely embedded F-space \(Y\) for which the operator restricted to \(Y\) is continuous and hypercyclic, i.e., the hypercyclicity of these operators cannot be a consequence of the comparison principle with hypercyclic operators on F-spaces.

### MSC:

47A16 | Cyclic vectors, hypercyclic and chaotic operators |

47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |

### Keywords:

hypercyclic operators; sequentially hypercyclic; composition operator; direct sums of sequence space; comparison principle
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\textit{A. Peris}, Funct. Approximatio, Comment. Math. 59, No. 2, 279--284 (2018; Zbl 1475.47010)

### References:

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