## 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.)

### Citations:

Zbl 1272.47037; Zbl 1150.47005
Full Text:

### References:

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