Strongly $n$-supercyclic operators

Abstract

Abstract. In this paper, we are interested in the properties of a new class of operators, recently introduced by Shkarin, called strongly n-supercyclic operators. This notion is stronger than n-supercyclicity. We prove that such operators have interesting spectral properties and give examples and counter-examples answering some natural questions asked by Shkarin. IntroductionIn what follows X will denote completely separable Baire vector spaces over the field K = R, C and T will be a bounded linear operator on X. Since the last 1980's, density properties of orbits of operators have been of great interest for many mathematicians, particularly hypercyclic and cyclic operators for their link with the invariant subspace problem. Another reason explaining this interest is that they appear in many well-known classes of operators: weighted shifts, composition operators, translation operators,...is dense in X. The set of all hypercyclic vectors for T is denoted by HC(T ). The operator T is said to be hypercyclic if HC(T ) = ∅.One may remove linearity in this definition, then under the same assumptions, T is said to be universal. In the same way, in 1974, Hilden and Wallen [10] introduced the weaker notion of supercyclicity which does not deal with orbits of vectors any more but with orbits of lines. Definition 1.2. A vector x ∈ X is said supercyclic for T if its projective orbit {λT n x, n ∈ N, λ ∈ K} is dense in X. The set of all hypercyclic vectors for T is denoted by SC(T ). The operator T is called hypercyclic if SC(T ) = ∅.As we said before, these properties have been intensively studied and the reader can refer to [2] and [9] for a deep and complete survey. One of the main ingredient providing such operators is the so called Hypercyclicity Criterion given by Carol Kitai in 1982 [11].Theorem: Hypercyclicity Criterion. Let X be a separable Banach space and T ∈ L(X). T satisfies the Hypercyclicity Criterion if there exist a strictly increasing sequence (n k ) k∈N , two dense setsIf T satisfies the Hypercyclicity Criterion, then T is hypercyclic. Shkarin [14] proved that this criterion is only a sufficient condition for hypercyclicity providing counter-examples to the necessary condition. Actually, J. Bès and A. Peris [4] showed that any finite direct sum of an operator T with itself is hypercyclic if and only if T satisfies the Hypercyclicity Criterion. This characterisation will be of great use later. Similarly, H.N. Salas [12] gave a Supercyclicity Criterion which is only a sufficient condition too and verifies the same kind of characterisation as above.Theorem: Supercyclicity Criterion. Let X be a separable Banach space and T ∈ L(X). T satisfies the Supercyclicity Criterion if there exist a strictly increasing sequence (n k ) k∈N , two dense sets D 1 , D 2 ⊂ X in X and a sequence of maps S n k : D 2 → X such that: (a) T n k x S n k y → 0 for any x ∈ D 1 and y ∈ D 2 ; (b) T n k S n k y → y for any y ∈ D 2 . If T satisfies the Supercyclicity Criterion, then T is supercyclic.These results are at the very heart of th...

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

“…This proves, in particular, that there exist n-supercyclic operators that are not strongly n-supercyclic and answers the question of the equivalence between n-supercyclicity and strong n-supercyclicity raised in [13]. The interested reader shall refer to [4] for other properties on strongly n-supercyclic operators in the infinite dimensional spaces setting.…”

“…which can be rewritten: According to (4), for every i ∈ N we have r − ε < |λ n i | < r + ε. Divide then (3) by n i λ n i :…”

“…However, in the real setting, we noticed in the previous section that n-supercyclicity can occur and thus the question for strong n-supercyclicity is still open. For this purpose we need the following proposition from [4]. It provides a more concrete definition of strongly n-supercyclic operators: Proposition 3.1.…”

“…It provides a more concrete definition of strongly n-supercyclic operators: Proposition 3.1. (Proposition 1.13 [4]) Let X be a completely separable Baire vector space. The following are equivalent:…”

“…Unfortunately, Shkarin did not go further in the study of strongly n-supercyclic operators. A study of general properties of strongly n-supercyclic operators can be found in [4].…”