• 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.

• 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.49 no.1
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• pp.1-6
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• 2009

Let B(X) denote the algebra of operators on a complex Banach space X, H(X) = {h ${\in}$ B(X) : h is hermitian}, and J(X) = {x ${\in}$ B(X) : x = $x_1$ + $ix_2$, $x_1$ and $x_2$ ${\in}$ H(X)}. Let ${\delta}_a$ ${\in}$ B(B(X)) denote the derivation ${\delta}_a$ = ax - xa. If J(X) is an algebra and ${\delta}_a^{-1}(0){\subseteq}{\delta}_{a^*}^{-1}(0)$ for some $a{\in}J(X)$, then ${\parallel}a{\parallel}{\leq}{\parallel}a-(x^*x-xx^*){\parallel}$ for all $x{\in}J(X){\cap}{\delta}_a^{-1}(0)$. The cases J(X) = B(H), the algebra of operators on a complex Hilbert space, and J(X) = $C_p$, the von Neumann-Schatten p-class, are considered.

• Kyungpook Mathematical Journal
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• v.48 no.2
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• pp.281-286
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• 2008

In this note we prove that if A and $B^*$ are subnormal operators and is a bounded linear operator such that AX - XB is a Hilbert-Schmidt operator, then f(A)X - Xf(B) is also a Hilbert-Schmidt operator and $${\parallel}f(A)X\;-\;Xf(B){\parallel}_2\;\leq\;L{\parallel}AX\;-\;XB{\parallel}_2$$, for f belonging to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and $X\;{\in}\;\cal{L}(\cal{H})$ is such that SX - XT belongs to a norm ideal (J, ${\parallel}\;{\cdot}\;{\parallel}_J$) and prove that f(S)X - Xf(T) $\in$ J and ${\parallel}f(S)X\;-\;Xf(T){\parallel}_J\;\leq\;C{\parallel}SX\;-\;XT{\parallel}_J$, for f in a certain class of functions.

• Kyungpook Mathematical Journal
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• v.48 no.1
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• pp.15-24
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• 2008

Let $N{\geq}1$ and p > 1. Let ${\Omega}$ be a domain of $\mathbb{R}^N$. In this article we shall establish Kato's inequalities for quasilinear degenerate elliptic operators of the form $A_pu$ = divA(x,$\nabla$u) for $u{\in}K_p({\Omega})$, ), where $K_p({\Omega})$ is an admissible class and $A(x,\xi)\;:\;{\Omega}{\times}\mathbb{R}^N{\rightarrow}\mathbb{R}^N$ is a mapping satisfying some structural conditions. If p = 2 for example, then we have $K_2({\Omega})\;= \;\{u\;{\in}\;L_{loc}^1({\Omega})\;:\;\partial_ju,\;\partial_{j,k}^2u\;{\in}\;L_{loc}^1({\Omega})\;for\;j,k\;=\;1,2,{\cdots},N\}$. Then we shall prove that $A_p{\mid}u{\mid}\;\geq$ (sgn u) $A_pu$ and $A_pu^+\;\geq\;(sgn^+u)^{p-1}\;A_pu$ in D'(${\Omega}$) with $u\;\in\;K_p({\Omega})$. These inequalities are called Kato's inequalities provided that p = 2. The class of operators $A_p$ contains the so-called p-harmonic operators $L_p\;=\;div(\mid{{\nabla}u{\mid}^{p-2}{\nabla}u)$ for $A(x,\xi)={\mid}\xi{\mid}^{p-2}\xi$.

• Journal of the Korean Mathematical Society
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• v.59 no.3
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• pp.469-494
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• 2022

In this paper, we study the boundedness of a class of inhomogeneous Journé's product singular integral operators on the inhomogeneous product Lipschitz spaces. The consideration of such inhomogeneous Journé's product singular integral operators is motivated by the study of the multi-parameter pseudo-differential operators. The key idea used here is to develop the Littlewood-Paley theory for the inhomogeneous product spaces which includes the characterization of a special inhomogeneous product Besov space and a density argument for the inhomogeneous product Lipschitz spaces in the weak sense.

• Journal of the Korean Mathematical Society
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• v.47 no.2
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• pp.235-246
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• 2010

A Banach space operator A $\in$ B(X) is polaroid if the isolated points of the spectrum of A are poles of the resolvent of A; A is hereditarily polaroid, A $\in$ ($\mathcal{H}\mathcal{P}$), if every part of A is polaroid. Let $X^n\;=\;\oplus^n_{t=i}X_i$, where $X_i$ are Banach spaces, and let A denote the class of upper triangular operators A = $(A_{ij})_{1{\leq}i,j{\leq}n$, $A_{ij}\;{\in}\;B(X_j,X_i)$ and $A_{ij}$ = 0 for i > j. We prove that operators A $\in$ A such that $A_{ii}$ for all $1{\leq}i{\leq}n$, and $A^*$ have the single-valued extension property have spectral properties remarkably close to those of Jordan operators of order n and n-normal operators. Operators A $\in$ A such that $A_{ii}$ $\in$ ($\mathcal{H}\mathcal{P}$) for all $1{\leq}i{\leq}n$ are polaroid and have SVEP; hence they satisfy Weyl's theorem. Furthermore, A+R satisfies Browder's theorem for all upper triangular operators R, such that $\oplus^n_{i=1}R_{ii}$ is a Riesz operator, which commutes with A.

• Journal of the Korean Mathematical Society
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• v.33 no.3
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• pp.575-585
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• 1996

Recently, existence theorems of coupled fixed points for mixed monotone operators have been considered by several authors (see [1]-[3], [6]). In this paper, we are continuously going to study the existence problems of coupled fixed points for two more general classes of mixed monotone operators. As an application, we utilize our main results to show thee existence of coupled fixed points for a class of non-linear integral equations.

• Journal of the Korean Mathematical Society
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• v.55 no.1
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• pp.225-251
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• 2018

The m-isometry of a single operator in Agler and Stankus [3] was naturally generalized to the m-isometric tuple of several commuting operators by Gleason and Richter [22]. Some examples of m-isometric tuples including the recently much studied Arveson-Drury d-shift were given in [22]. We provide more examples of m-isometric tuples of operators by using sums of operators or products of operators or functions of operators. A class of m-isometric tuples of unilateral weighted shifts parametrized by polynomials are also constructed. The examples in Gleason and Richter [22] are then obtained by choosing some specific polynomials. This work extends partially results obtained in several recent papers on the m-isometry of a single operator.

• Bulletin of the Korean Mathematical Society
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• v.51 no.5
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• pp.1241-1257
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• 2014

We consider a class of hypoelliptic operators of the following type $$L=\sum_{i,j=1}^{p_0}a_{ij}{\partial}^2_{x_ix_j}+\sum_{i,j=1}^{N}b_{ij}x_i{\partial}_{x_j}-{\partial}_t$$, where ($a_{ij}$), ($b_{ij}$) are constant matrices and ($a_{ij}$) is symmetric positive definite on $\mathbb{R}^{p_0}$ ($p_0{\leqslant}N$). By establishing global Morrey estimates of singular integral on the homogenous space and the relation between Morrey space and weak Morrey space, we obtain the global weak Morrey estimates of the operator L on the whole space $\mathbb{R}^{N+1}$.