Operator on the space of rapidly decreasing functions with all non-zero vectors hypercyclic

Invariant subspaces for non-normable Fréchet spaces

“…We then look at the approach of Goliński concerning the existence of operators without non-trivial invariant subspaces. In his paper [10], Goliński states sufficient conditions for the construction of a Read-type operator without non-trivial invariant subspaces and he shows in [11] that it is also possible to construct a Read-type operator without non-trivial invariant subset on s. In this paper, we state sufficient conditions for the construction of a Read-type operator without non-trivial invariant subset. While the conditions stated by Goliński concerned only Köthe sequence spaces of type ℓ 1 , our results can be applied to any Fréchet space with a Schauder basis and thus in particular to any Köthe sequence space.…”

“…On the other hand, if X is a Fréchet space without continuous norm satisfying the conditions of Theorem 2.1 then ker p j0 is isomorphic to ω (see [19,Proposition 26.16]) and thus X is isomorphic to ω ' Y where Y is a Fréchet space with continuous norm. Thanks to Atzmon [3,4] and Goliński [10,11], we know several examples of non-normable Fréchet spaces which do not satisfy the Invariant Subset Property or the Invariant Subspace Property. In the next section, we investigate sufficient conditions for having no non-trivial invariant subset by trying to take the best advantage of the non-normability of non-normable Fréchet spaces.…”

“…We start by fixing the family of polynomials Q n,k . The choice of this family will follow from the following lemma (see for instance [11,Lemma 3]). Lemma 4.5.…”

“…In 2012, Goliński [10] exhibited classical examples by showing that the space of entire functions HpCq and the Schwartz space of rapidly decreasing functions s do not satisfy the Invariant Subspace Property. He even showed that the space s does not satisfy the Invariant Subset Property [11]. However, it is not known if the space HpCq does not satisfy the Invariant Subset Property.…”

The invariant subspace problem for the space of smooth functions on the real line

“…Now the equation (7) gives as the values for α j for j ∈ [pos(∆ n , 0), pos(a n , 0)). Hence the values of T j e 0 are defined for j < pos(a n , 0) + pos(∆ n , 0) by (6). We put ∆ n+1 = a n + pos(∆ n , 0).…”