UNIVERSALITY OF AFFINE SEMI-GROUPS ON SUPERCYCLICITY OF THE SEQUENCE OPERATORS

In this paper, show that for all supercyclic strongly continuous sequence of operators semi-group on a complex F-space, every T(1+ε) j with ε > −1 is supercyclic. Also, the sets of all supercyclic vectors T(1+ε) j with ε > −1 are precisely the sets of supercyclic vectors of the entire semi-group.

[1] and references therein have been covered the concept of Hypercyclicity and super-cyclicity, also discuss the relation between the supercyclicity of a linear semi-group and supercyclicity of the individual members of the semi-group. The hypercyclicity version of the question was treated by Conejero, Müller and Peris [2], who proved that for every strongly continuous hypercyclic linear semi-groups Virtually the same proof works in the following much more general setting. To show that in the case = ℂ, the supercyclicity version of Theorem A holds without any additional assumptions We can applying Proposition 1.1. with the same results in [9] and considering the induced action on subsets of the projective space. In the other case the topic reduces to the generalization of Theorem A to affine semi-groups see e.g.e., [4].

A DICHOTOMY FOR SUPERCYCLIC LINEAR SEMI-GROUPS
An analogue of the following result for individual supercyclic sequence of operators. nowhere dense. Since is closed, its interior is non-empty. Since is closed and 0 ∉ , we can find a non-empty balanced open set such that ∩ = ∅. Clearly ∈ whenever ∈ and ∈ , ≠ 0. Since is open and together with the latter property of implies that the open set = ∩ is non-empty. Taking into account the definition of , the inclusion ⊆ , the equality ∩ = ∅ and the fact that is balanced, see that every ∈ can be written as = , where ∈ and ∈ = { ∈ : | | ≤ 1}.
Since both and are compact, = { : ∈ , ∈ }is a compact subset of . Since ⊆ , is a non-empty open set with compact closure. Such a set exists [5] only if is finite dimensional. This contradiction completes the proof.
The following lemma is a particular case in [6].
therefore is not uphold supercyclic strongly continuous linear semi-groups. Proof. Using the semi-group property, it is easy to see that is invariant for all (1+ ) .

REMARKS
The following example shows that the hypercyclicity of the underlying linear semi-group is not implies universality of a strongly continuous affine semi-group see e.g. [4]. + . This however is not the case as shown in [9].