• The Pure and Applied Mathematics
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• v.7 no.2
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• pp.79-86
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• 2000

In this paper, we introduce the classes H(p,q,k),K(p;k) of operators determined by the Heinz-Kato-Furuta inequality and Holer-McCarthy inequality. We characterize relationship between p-quasihyponormal, $\kappa$-quasihyponormal and $\kappa$-p-quasihyponormal operators. And it is proved that every operator in K(p;1) for some $0 is paranormal.

• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.63-71
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• 2015

Let H be a separable infinite dimensional complex Hilbert space, and let $\mathcal{L}(H)$ denote the algebra of all bounded linear operators on H into itself. Let $A,B{\in}\mathcal{L}(H)$ we define the generalized derivation ${\delta}_{A,B}:\mathcal{L}(H){\mapsto}\mathcal{L}(H)$ by ${\delta}_{A,B}(X)=AX-XB$, we note ${\delta}_{A,A}={\delta}_A$. If the inequality ${\parallel}T-(AX-XA){\parallel}{\geq}{\parallel}T{\parallel}$ holds for all $X{\in}\mathcal{L}(H)$ and for all $T{\in}ker{\delta}_A$, then we say that the range of ${\delta}_A$ is orthogonal to the kernel of ${\delta}_A$ in the sense of Birkhoff. The operator $A{\in}\mathcal{L}(H)$ is said to be finite [22] if ${\parallel}I-(AX-XA){\parallel}{\geq}1(*)$ for all $X{\in}\mathcal{L}(H)$, where I is the identity operator. The well-known inequality (*), due to J. P. Williams [22] is the starting point of the topic of commutator approximation (a topic which has its roots in quantum theory [23]). In [16], the author showed that a paranormal operator is finite. In this paper we present some new classes of finite operators containing the class of paranormal operators and we prove that the range of a generalized derivation is orthogonal to its kernel for a large class of operators containing the class of normal operators.

• 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.

• Communications of the Korean Mathematical Society
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• v.39 no.2
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• pp.437-460
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• 2024

In this paper we present the weighted shift operators having the property of moment infinite divisibility. We first review the monotone theory and conditional positive definiteness. Next, we study the infinite divisibility of sequences. A sequence of real numbers γ is said to be infinitely divisible if for any p > 0, the sequence γ^p = {γ^p_n}^∞_n=0 is positive definite. For sequences α = {α_n}^∞_n=0 of positive real numbers, we consider the weighted shift operators W_α. It is also known that W_α is moment infinitely divisible if and only if the sequences {γ_n}^∞_n=0 and {γ_n+1}^∞_n=0 of W_α are infinitely divisible. Here γ is the moment sequence associated with α. We use conditional positive definiteness to establish a new criterion for moment infinite divisibility of W_α, which only requires infinite divisibility of the sequence {γ_n}^∞_n=0. Finally, we consider some examples and properties of weighted shift operators having the property of (k, 0)-CPD; that is, the moment matrix M_γ(n, k) is CPD for any n ≥ 0.