• 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.

• 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.

• 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.

• 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.