• Journal of Integrative Natural Science
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• v.10 no.2
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• pp.115-118
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• 2017

We study a notion of $\mathcal{A}$-frequent hypercyclicity of linear maps between the Banach algebras consisting of operators on a separable infinite dimensional Banach space. We prove a sufficient condition for a linear map to satisfy the $\mathcal{A}$-frequent hypercyclicity in the strong operator topology.

• Bulletin of the Korean Mathematical Society
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• v.60 no.2
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• pp.293-305
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• 2023

Let {T_t}_t∈∆ be the translation semigroup with a sector ∆ ⊂ ℂ as index set. The recurrent hypercyclicity criterion (RHCC) for the C₀-semigroup {T_t}_t∈∆ is established, and then the equivalent conditions ensuring {T_t}_t∈∆ satisfying the RHCC on weighted spaces of p-integrable and of continuous functions are presented. Especially, every chaotic semigroup {T_t}_t∈∆ satisfies the RHCC.

• Journal of the Korean Mathematical Society
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• v.43 no.6
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• pp.1219-1229
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• 2006

Let X be a separable Banach space. We give sufficient conditions under which $T:X{\rightarrow}X$ is hereditarily hypercyclic. Also, we prove that hereditarily hypercyclicity with respect to a special sequence implies the hereditarily hypercyclicity with respect to the entire sequence.

• Bulletin of the Korean Mathematical Society
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• v.44 no.1
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• pp.117-123
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• 2007

In this paper, we first adapt Runge's Theorem to work on certain domains in any complex Banach space. Then, using this result, we extend Birkhoff's Theorem on the hypercyclicity of translations on $H(\mathbb{C})$ and Costakis' and Sambarino's result on the existence of common hypercyclic functions for uncountable families of translations on $H(\mathbb{C})$ to subs paces of $H_b(E)$ (in some cases all of $H_{b}$(E)), E being in a large class of Banach spaces.

• 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.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1269-1283
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• 2015

An operator $T{{\in}}{\mathcal{L}}({\mathcal{H}})$ is said to be skew complex symmetric if there exists a conjugation C on ${\mathcal{H}}$ such that $T=-CT^*C$. In this paper, we study properties of skew complex symmetric operators including spectral connections, Fredholmness, and subspace-hypercyclicity between skew complex symmetric operators and their adjoints. Moreover, we consider Weyl type theorems and Browder type theorems for skew complex symmetric operators.

• Journal of the Korean Mathematical Society
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• v.51 no.2
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• pp.363-382
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• 2014

In this paper we first characterize the hereditarily hypercyclicity of the unilateral (or bilateral) weighted shifts on the spaces $L^2(\mathbb{N},\mathcal{K})$ (or $L^2(\mathbb{Z},\mathcal{K})$) with weight sequence {$A_n$} of positive invertible diagonal operators on a separable complex Hilbert space $\mathcal{K}$. Then we give the necessary and sufficient conditions for the supercyclicity of those weighted shifts, which extends some previous results of H. Salas. At last, we give some conditions for the supercyclicity of three different weighted shifts.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.557-563
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• 2019

In this paper we provide a $Ces{\grave{a}}ro$-hypercyclicity criterion and offer two examples of this criterion. At the same time, we also characterize other properties of $Ces{\grave{a}}ro$-hypercyclic operators.

• 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 $H({\mathbb{C}}^d)$.