Some properties of N-supercyclic operators

Abstract

Abstract. Let T be a continuous linear operator on a Hausdorff topological vector space X over the field C. We show that if T is N -supercyclic, i.e., if X has an N -dimensional subspace whose orbit under T is dense in X , then T * has at most N eigenvalues (counting geometric multiplicity). We then show that N -supercyclicity cannot occur nontrivially in the finite-dimensional setting: the orbit of an N -dimensional subspace cannot be dense in an (N + 1)-dimensional space. Finally, we show that a subnormal operator on an infinite-dimensional Hilbert space can never be N -supercyclic.

Hyponormal Operators, Weighted Shifts and Weak Forms of Supercyclicity

“…It is not known whether or not Ansari's theorem is still valid in the N -supercyclic setting; the problem is explicitly raised in [3]. The following lemma is a kind of substitute, which may help to solve it.…”

“…Questions 1.1 and 1.2 are reminiscent of two results of Kitai [11] (in the hypercyclic case) and Bourdon [2]: no hyponormal operator can be supercyclic. Question 1.3 seems interesting because N -supercyclic operators which appear in [6] or in [3] are constructed as direct sums of supercyclic operators, and it would be nice to obtain some other, less 'ad hoc' examples. Moreover, hypercyclic and supercyclic weighted shifts have already been characterized by Salas [15,16], so it is natural to consider N -supercyclic case.…”

“…A more successful generalization of supercyclicity is the notion of N -supercyclicity, which was introduced in [6] and also studied in [3]. Observe that an operator T ∈ L(X) is supercyclic if and only if there exists a one-dimensional subspace of X whose orbit under T is dense in X.…”

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

“…We progress step by step considering particular cases until we reach the remaining part of Theorem 1 in the general case. We begin by proving two special cases: the case of a real Jordan block matrix of size 2 is considered first because it is the simplest matrix that Bourdon, Feldman and Shapiro have not checked in [3] and then the case of a direct sum of rotation matrices because it permits to notice that something more is needed if one wants to go further. These two results are stated and proved first because their proofs introduce some techniques involved for more general proofs.…”

“…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:…”