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

Ernst, Romuald

doi:10.4064/sm220-1-2

Cited by 2 publications

(7 citation statements)

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“…Indeed, this would solve the Ansari property problem for n supercyclic operators. In fact, the present author gave a negative answer to this question [7] and we will construct some more counterexamples in the present paper. Since, [13] is very concise on strongly n-supercyclicity giving only the definition and the Ansari property and [7] is only concerned with the finite dimensional setting, the aim of this paper is to present a complete study of strong n-supercyclicity.…”

Section: Definition 12mentioning

confidence: 85%

“…Recently, the present author [7] proved that things were different in the real setting providing the following theorem:…”

Section: Introductionmentioning

confidence: 94%

“…There is no (n − 1)-supercyclic operator on C n . In particular, there is no k-supercyclic operator on C n for any 1 ≤ k ≤ n − 1.Recently, the present author [7] proved that things were different in the real setting providing the following theorem:. Let n ≥ 2.…”

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confidence: 89%

“…It has been proved in [5] and [7] that strong n-supercyclicity is purely infinite dimensional. We are going to make use of this fact to construct an operator being strongly k-supercyclic if and only if k ≥ n. …”

Section: 1mentioning

confidence: 99%

“…He defines: Definition 1.3. An operator T is said to be n-supercyclic, n ≥ 1, if there is a subspace of dimension n in X with dense orbit.These operators have been studied in [1], [3] and [5] and [7]. Feldman gave some different classes of n-supercyclic operators and in particular:…”

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confidence: 99%

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Strongly $n$-supercyclic operators

Ernst¹

2014

J. Operator Theory

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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...

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Section: Definition 12mentioning

confidence: 85%

“…Recently, the present author [7] proved that things were different in the real setting providing the following theorem:…”

Section: Introductionmentioning

confidence: 94%

mentioning

confidence: 89%

Section: 1mentioning

confidence: 99%

mentioning

confidence: 99%