• Kyungpook Mathematical Journal
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• v.50 no.1
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• pp.89-100
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• 2010

Let g = (V, E, ${\mu}$) be a weighted directed tree, where V is a vertex set, E is an edge set, and ${\mu}$ is ${\sigma}$-finite measure on V. The tree g induces a composition operator C on the Hilbert space $l^2$(V). Hand-type directed trees are defined and characterized the weak hyponormalities of such C in this note. Also some additional related properties are discussed. In addition, some examples related to directed hand-type trees are provided to separate classes of weak-hyponormal operators.

• 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.

• Kyungpook Mathematical Journal
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• v.58 no.1
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• pp.81-90
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• 2018

A bounded linear operator T on a Hilbert space ${\mathcal{H}}$ is convex, if for each $x{\in}{\mathcal{H}}$, ${\parallel}T^2x{\parallel}^2-2{\parallel}Tx{\parallel}^2+{\parallel}x{\parallel}^2{\geq}0$. In this paper, it is shown that if T is convex and supercyclic then it is a contraction or an expansion. We then present some examples of convex supercyclic operators. Also, it is proved that no convex composition operator induced by an automorphism of the disc on a weighted Hardy space is supercyclic.