n-supercyclic and strongly n-supercyclic operators in finite dimensions

We prove that on R , there is no n-supercyclic operator with 1 ≤ n < b 2 c i.e. if R has an n-dimensional subspace whose orbit under T ∈ L(R ) is dense in R , then n is greater than b 2 c. Moreover, this value is optimal. We then consider the case of strongly n-supercyclic operators. An operator T ∈ L(R ) is strongly n-supercyclic if R has an ndimensional subspace whose orbit under T is dense in Pn(R ), the n-th Grassmannian. We prove that strong n-supercyclicity does not occur non-trivially in finite dimension. Let T be a continuous linear operator on a Banach space X. The orbit of a set E ∈ X under T is defined by O(E, T ) := ∪n∈Z+T(E). Many authors have already studied some density properties of such orbits for different original sets E. If E is a singleton and O(E, T ) is dense in X, then T is said to be hypercyclic. Hypercyclicity has been first studied by Birkhoff in 1929 and has been a subject of great interest during the last twenty years, see [2] and [6] for a survey on hypercyclicity. Later, in 1974, Hilden and Wallen [9] worked on a different set E = Kx which is a one-dimensional subspace of X, and if O(E, T ) is dense in X, then T is said to be supercyclic. Several generalisations of supercyclicity were proposed since like the one introduced by Feldman [5] in 2002. Rather than considering orbits of lines, Feldman defines an n-supercyclic operator as being an operator for which there exists an n-dimensional subspace E such that O(E, T ) is dense in X. This notion has been mainly studied in [1], [3] and [5]. In 2004, Bourdon, Feldman and Shapiro proved in the complex case that non-trivial n-supercyclicity is purely infinite dimensional: Theorem Bourdon, Feldman, Shapiro. Let N ≥ 2. Then there is no (N − 1)-supercyclic operator on CN . In particular, there is no n-supercyclic operator on CN for 1 ≤ n ≤ N − 1. The last theorem extends a result proved by Hilden and Wallen for supercyclic operators in the complex setting. On the other hand, Herzog proved that there is no supercyclic operators on Rn for n ≥ 3 in [8]. Therefore, it is natural to ask the question of the existence of n-supercyclic operators in the real setting. In 2008, Shkarin introduced another generalisation of supercyclicity in [13]. Roughly speaking, an operator T ∈ L(X) is strongly n-supercyclic if there exists a subspace of dimension n whose orbit is dense in the set of n-dimensional subspaces of X. To be more precise, we need to define the topology of this set, which is called the n-th-Grassmannian of X. If dim(X) ≥ n then one may define a topology on the n-th Grassmannian. To do this, let us consider the open subset Xn of all linearly independent n-tuples with the topology induced from X n and let πn : Xn → Pn(X) be defined by πn(x) = span{x1, . . . , xn}. The topology on Pn(X) is the coarsest topology for which the map πn is open and continuous. Let us now turn to the definition of strong n-supercyclicity: M ∈ Pn(X) is a strongly n-supercyclic subspace for T if every T k(M) is n-dimensional and if {T k(M), k ∈ Z+} is dense in Pn(X). If such a subspace exists, then T is said to be strongly n-supercyclic. We denote by ESn(T ) the set of strongly n-supercyclic subspaces for an operator T . An open question regarding n-supercyclic operators is to know whether they satisfy the Ansari property: is it true that T p is n-supercyclic for any p ≥ 2 provided T itself is n-supercyclic? Shkarin [13] has shown that strongly n-supercyclic operators do satisfy the Ansari property and 1