Operators Commuting with the Volterra Operator are not Weakly Supercyclic

Shkarin, Stanislav

doi:10.1007/s00020-010-1790-y

Cited by 8 publications

(5 citation statements)

References 30 publications

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“…Now we can prove Proposition 1.4. Its proof resembles the proof of the main result in [14] and gives an idea of the proof of Theorem 1.1 in the following sections. For f ∈ L 1 [0, 1], we say that the infimum of the support of f is 0 if for every ε > 0, f does not vanish (almost everywhere) on [0, ε].…”

Section: Obstacles To Weak Supercyclicitymentioning

confidence: 90%

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Orbits of Semigroups of Truncated Convolution Operators

Shkarin¹

2012

Glasgow Math. J.

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Abstract. We prove that a semigroup generated by finitely many truncated convolution operators on L p [0, 1] with 1 p < ∞ is non-supercyclic. On the other hand, there is a truncated convolution operator, which possesses irregular vectors.2010 Mathematics Subject Classification. 47A16, 37B99. Introduction.Throughout the paper, all vector spaces are assumed to be over the field ‫ދ‬ being either the field ‫ރ‬ of complex numbers or the field ‫ޒ‬ of real numbers, ‫ޚ‬ is the set of integers, ‫ޚ‬ + is the set of non-negative integers, ‫ޒ‬ + is the set of non-negative real numbers and ‫ގ‬ is the set of positive integers. The symbol L(X) stands for the space of continuous linear operators on a topological vector space X and X * is the space of continuous linear functionals on X. A family F = {F a : a ∈ A} of continuous maps from a topological space X to a topological space Y is called universal if there is x ∈ X for which the orbit O(F, x) = {F a x : a ∈ A} is dense in Y . Such an x is called a universal element for F. We use the symbol U(F) to denote the set of universal elements for F. If X is a topological vector space, and F is a commutative subsemigroup of L(X), then we call F hypercyclic if F is universal and the members of U(F) are called hypercyclic vectors for F. We say that F is supercyclic if the semigroup F p = {zT : z ∈ ‫,ދ‬ T ∈ F} is hypercyclic and the hypercyclic vectors for F p are called supercyclic vectors for F. An orbit O(F p , x) will be called a projective orbit of F. We refer to a tuple T = (T 1 , . . . , T n ) of commuting continuous linear operators on X as hypercyclic (respectively, supercyclic) if the semigroup generated by T is hypercyclic (respectively, supercyclic). The concept of hypercyclic tuples of operators was introduced and studied by Feldman [5]. In the case n = 1, it becomes the conventional hypercyclicity (or supercyclicity), which has been widely studied, see the book [1] and references therein.Gallardo-Gutiérrez and Montes-Rodríguez [6], answering a question of Salas, proved that the Volterra operator

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Section: Obstacles To Weak Supercyclicitymentioning

confidence: 90%

“…Léon and Piqueras [7] raised a question whether any T ∈ L(L p [0, 1]) commuting with V is not weakly supercyclic. This question was answered affirmatively in [14]. Still there remained a possibility of existence of a hypercyclic or at least supercyclic tuple of truncated convolution operators.…”

Section: Introductionmentioning

confidence: 98%

Orbits of Semigroups of Truncated Convolution Operators

Shkarin¹

2012

Glasgow Math. J.

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“…Since a Banach space of finite dimension > 1 supports no supercyclic operators (see [12]), a supercyclic Banach algebra of dimension > 1 must be infinite dimensional. According to [10,Proposition 3.4], an infinite dimensional commutative Banach algebra B is radical if there is b ∈ B for which the multiplication operator M b is supercyclic. Since a supercyclic Banach algebra of dimension > 1 is infinite dimensional, commutative and has a supercyclic multiplication operator, A is radical.…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…There is no point to consider 'hypercyclic Banach algebras' in the obvious sense. Indeed, in [10] it is observed that a multiplication operator on a commutative Banach algebra is never hypercyclic. Obviously, supercyclic as well as almost hypercyclic Banach algebras are commutative and separable.…”

Section: Introductionmentioning

confidence: 99%

Chaotic Banach algebras

Shkarin

2010

Preprint

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“…Non-supercyclicity of the Volterra operator on L p [0, 1], 1 ≤ p < ∞ is proved in [9]. Later, Shkarin [16] generalized this result, and proved that operators commuting with the Volterra operator on such spaces are not weakly supercyclic. In this paper, we are going to prove that operators in the commutant of a cyclic convolution operator on the Hardy space are not weakly supercyclic.…”

Section: Introductionmentioning

confidence: 99%