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The following open problem is stated: Is there a non-normable Fréchet space E such that every continuous linear operator T on E has the form T = λI + S, where S maps a 0-neighbourhood of E into a bounded set? A few remarks and the relation of this question with other still open problems on operators between Fréchet spaces are mentioned.
In this paper we review some known characterizations of the weak mixing property for operators on topological vector spaces, extend some of them, and obtain new ones.
We give sufficient conditions for chaos of (differential) operators on Hilbert spaces of entire functions. To this aim we establish conditions on the coefficients of a polynomial P(z) such that P(B) is chaotic on the space $${\ell^p}$$ , where B is the backward shift operator.
We unify and extend several results on the dynamic behaviour of composition operators on the space of holomorphic functions on a simply connected plane domain and endowed with the compact open topology. In particular, we show that a composition operator is weakly supercyclic if and only if the algebra it generates consists entirely, except for the null one, of operators that are topologically mixing,...
We show that the non-zero multiples of the derivative operator and the non-zero multiples of non-trivial translation operators on the space of entire functions share a common hypercyclic subspace, i.e. a closed infinite-dimensional subspace in which each non-zero vector has a dense orbit for each of these operators.
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