• Title, Summary, Keyword: hypercyclic operator

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q-FREQUENT HYPERCYCLICITY IN AN ALGEBRA OF OPERATORS

  • Heo, Jaeseong;Kim, Eunsang;Kim, Seong Wook
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.443-454
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    • 2017
  • We study a notion of q-frequent hypercyclicity of linear maps between the Banach algebras consisting of operators on a separable infinite dimensional Banach space. We derive a sufficient condition for a linear map to be q-frequently hypercyclic in the strong operator topology. Some properties are investigated regarding q-frequently hypercyclic subspaces as shown in [5], [6] and [7]. Finally, we study q-frequent hypercyclicity of tensor products and direct sums of operators.

HYPERCYCLIC OPERATOR WEIGHTED SHIFTS

  • Hazarika, Munmun;Arora, S.C.
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.4
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    • pp.589-598
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    • 2004
  • We consider bilateral operator weighted shift T on $L^2$(K) with weight sequence ${[A_{n}]_{n=-{\infty}}}^{\infty}$ of positive invertible diagonal operators on K. We give a characterization for T to be hypercyclic, and show that the conditions are far simplified in case T is invertible.

SYNDETIC SEQUENCES AND DYNAMICS OF OPERATORS

  • Rezaei, Hamid
    • Communications of the Korean Mathematical Society
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    • v.27 no.3
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    • pp.537-545
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    • 2012
  • In the present paper, we show that a continuous linear operator T on a Frechet space satisfies the Hypercyclic Criterion with respect to a syndetic sequence must satisfy the Kitai Criterion. On the other hand, an operator, hereditarily hypercyclic with respect to a syndetic sequence must be mixing. We also construct weighted shift operators satisfying the Hypercyclicity Criterion which do not satisfy the Kitai Criterion. In other words, hereditarily hypercyclic operators without being mixing.

HYPERCYCLICITY ON INVARIANT SUBSPACES

  • Petersson, Henrik
    • Journal of the Korean Mathematical Society
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    • v.45 no.4
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    • pp.903-921
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    • 2008
  • A continuous linear operator $T\;:\;X{\rightarrow}X$ is called hypercyclic if there exists an $x\;{\in}\;X$ such that the orbit ${T^nx}_{n{\geq}0}$ is dense. We consider the problem: given an operator $T\;:\;X{\rightarrow}X$, hypercyclic or not, is the restriction $T|y$ to some closed invariant subspace $y{\subset}X$ hypercyclic? In particular, it is well-known that any non-constant partial differential operator p(D) on $H({\mathbb{C}}^d)$ (entire functions) is hypercyclic. Now, if q(D) is another such operator, p(D) maps ker q(D) invariantly (by commutativity), and we obtain a necessary and sufficient condition on p and q in order that the restriction p(D) : ker q(D) $\rightarrow$ ker q(D) is hypercyclic. We also study hypercyclicity for other types of operators on subspaces of $H({\mathbb{C}}^d)$.

HYPERCYCLICITY OF WEIGHTED COMPOSITION OPERATORS ON THE UNIT BALL OF ℂN

  • Chen, Ren-Yu;Zhou, Ze-Hua
    • Journal of the Korean Mathematical Society
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    • v.48 no.5
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    • pp.969-984
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    • 2011
  • This paper discusses the hypercyclicity of weighted composition operators acting on the space of holomorphic functions on the open unit ball $B_N$ of $\mathbb{C}^N$. Several analytic properties of linear fractional self-maps of $B_N$ are given. According to these properties, a few necessary conditions for a weighted composition operator to be hypercyclic in the space of holomorphic functions are proved. Besides, the hypercyclicity of adjoint of weighted composition operators are studied in this paper.

STRONG HYPERCYCLICITY OF BANACH SPACE OPERATORS

  • Ansari, Mohammad;Hedayatian, Karim;Khani-Robati, Bahram
    • Journal of the Korean Mathematical Society
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    • v.58 no.1
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    • pp.91-107
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    • 2021
  • A bounded linear operator T on a separable infinite dimensional Banach space X is called strongly hypercyclic if $$X{\backslash}\{0\}{\subseteq}{\bigcup_{n=0}^{\infty}}T^n(U)$$ for all nonempty open sets U ⊆ X. We show that if T is strongly hypercyclic, then so are Tn and cT for every n ≥ 2 and each unimodular complex number c. These results are similar to the well known Ansari and León-Müller theorems for hypercyclic operators. We give some results concerning multiplication operators and weighted composition operators. We also present a result about the invariant subset problem.

ON SOME PROPERTIES OF J-CLASS OPERATORS

  • Asadipour, Meysam;Yousefi, Bahmann
    • Communications of the Korean Mathematical Society
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    • v.34 no.1
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    • pp.145-154
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    • 2019
  • The notion of hypercyclicity was localized by J-sets and in this paper, we will investigate for an equivalent condition through the use of open sets. Also, we will give a J-class criterion, that gives conditions under which an operator belongs to the J-class of operators.