Hyponormal Operators, Weighted Shifts and Weak Forms of Supercyclicity

Bayart, Frédéric; Matheron, Étienne

doi:10.1017/s0013091504000975

Cited by 38 publications

(44 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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:…”

mentioning

confidence: 99%

Strongly $n$-supercyclic operators

Ernst¹

2014

J. Operator Theory

View full text Add to dashboard Cite

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...

show abstract

mentioning

confidence: 99%

Strongly $n$-supercyclic operators

Ernst¹

2014

J. Operator Theory

View full text Add to dashboard Cite

show abstract

“…Feldman showed that normal operators cannot be N -supercyclic [11]. Later, Bayart and Matheron generalized this result to hyponormal operators [5]. Non-supercyclicity of m-isometric operators on Hilbert spaces has been shown by Faghih-Ahmadi and Hedayatian in [10].…”

Section: Let H Denote An Infinite Dimensional Hilbert Space and B(h) mentioning

confidence: 99%

N-SUPERCYCLICITY OF AN A-m-ISOMETRY

Hedayatian¹

2015

Honam Mathematical Journal

View full text Add to dashboard Cite

show abstract

“…The next example shows that such operators do exist. It is a modification of an example constructed in [2]. By T we denote the unit circle in C: T = {z ∈ C : |z| = 1}.…”

Section: Then T Is Hypercyclic [T M ] Has Dense Range and [T [T Mmentioning

confidence: 99%

Operators Commuting with the Volterra Operator are not Weakly Supercyclic

Shkarin

2010

Integr. Equ. Oper. Theory

View full text Add to dashboard Cite

We prove that any bounded linear operator on L p [0, 1] for 1 p < ∞, commuting with the Volterra operator V , is not weakly supercyclic, which answers affirmatively a question raised by Léon-Saavedra and Piqueras-Lerena. It is achieved by providing an algebraic flavored condition on an operator which prevents it from being weakly supercyclic and is satisfied for any operator commuting with V . MSC: 47A16, 37A25

show abstract

Hyponormal Operators, Weighted Shifts and Weak Forms of Supercyclicity

Cited by 38 publications

References 14 publications

Strongly $n$-supercyclic operators

Strongly $n$-supercyclic operators

N-SUPERCYCLICITY OF AN A-m-ISOMETRY

Operators Commuting with the Volterra Operator are not Weakly Supercyclic

Contact Info

Product

Resources

About