Diskcyclicity of Sets of Operators and Applications
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Diskcyclicity of Sets of Operators and Applications Mohamed AMOUCH
Otmane BENCHIHEB1)
University Chouaib Doukkali, Department of Mathematics, Faculty of science Eljadida, Morocco E-mail : [email protected] [email protected] Abstract In this paper, we introduce and study the diskcyclicity and disk transitivity of a set of operators. We establish a diskcyclicity criterion and give the relationship between this criterion and the diskcyclicity. As applications, we study the diskcyclicty of C0 -semigroups and C-regularized groups. We show that a diskcyclic C0 -semigroup exists on a complex topological vector space X if and only if dim(X) = 1 or dim(X) = ∞ and we prove that diskcyclicity and disk transitivity of C0 -semigroups (resp C-regularized groups) are equivalent. Keywords Hypercyclicity, supercyclicity, diskcyclicity, C0 -semigroups of operators, C-regularized groups of operators MR(2010) Subject Classification
1
47A16
Introduction and Preliminary
Let X be a complex topological vector space and B(X) the space of all continuous linear operators on X. By an operator, we always mean a continuous linear operator. The most studied notion in linear dynamics is that of hypercyclicity: An operator T is called hypercyclic if there is some vector x ∈ X such that the orbit of x under T ; Orb(T, x) = {T n x : n ∈ N}, is a dense subset of X, such a vector x is called a hypercyclic vector for T . The set of all hypercyclic vectors for T is denoted by HC(T ). The first example of hypercyclic operator was given by Rolewicz in [17]. He proved that if B is a backward shift on the Banach space p (N), then λB is a hypercyclic operator for any complex number λ satisfying |λ| > 1. Another important notion in dynamical system is that of supercyclicity. This notion was introduced by Hilden and Wallen in [12]. An operator T is called supercyclic if there is a vector x ∈ X such that the projective orbit of x under T ; COrb(T, x) = {λT n x : λ ∈ C, n ∈ N}, is dense in X. In this case, the vector x is said to be a supercyclic vector for T . The set of all supercyclic vectors for T is denoted by SC(T ). For more detailed information on both hypercyclicity and supercyclicity, the reader may refer to [7, 11]. Received July 21, 2019, revised November 9, 2019, accepted January 13, 2020 1) Corresponding author
Amouch M. and Benchiheb O.
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In the same spirit, since the operator λB is not hypercyclic whenever |λ| ≤ 1, we are motivated to study the disk orbit. An operator T is said to be diskcyclic if there is some vector x ∈ X such that the disk orbit of x under T ; DOrb(T, x) = {αT n x : α ∈ D, n ≥ 0}, is dense in X. Such vector x is called a diskcyclic vector for T , and the set of all diskcyclic vectors for T is denoted by DC(T ). An equivalent notion of the diskcyclicity in the case of a second countable complex topological vector space is that of disk transitivity. An operator T is said to be disk transitive if for
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