• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.1003-1008
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• 2010

In allelic model $X\;=\;(x_1,\;x_2,\;\cdots,\;x_d)$, $$M_f(t)\;=\;f(p(t))\;-\;\int_0^t\;Lf(p(t))ds$$ is a P-martingale for diffusion operator L under the certain conditions. In this note, we define $T_tf\;=\;E_{p_0}^{p^*}\;[f((P(t))]$ for $t\;{\geq}\;0$ for using a new diffusion operator $L^*$ and we show the diffusion relations between $T_t$ and diffusion operator $L^*$.

• Bulletin of the Korean Mathematical Society
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• v.42 no.3
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• pp.571-577
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• 2005

In this paper we prove that if T is a k-th root of a p­hyponormal operator when T is compact or T$^{n}$ is normal for some integer n > k, then T is (generalized) scalar, and that if T is a k-th root of a semi-hyponormal operator and have the property $\sigma$(T) is contained in an angle < 2$\pi$/k with vertex in the origin, then T is subscalar.

• Bulletin of the Korean Mathematical Society
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• v.42 no.3
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• pp.543-554
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• 2005

In this paper it is proved that for p-hyponormal or log-hyponormal operator A there exist an associated hyponormal operator T, a quasi-affinity X and an injection operator Y such that TX = XA and AY = YT. The operator A and T have the same spectral picture. We apply these results to give brief proofs of some well known spectral properties of p-hyponormal and log­hyponormal operators, amongst them that the spectrum is a con­tinuous function on these classes of operators.

• East Asian mathematical journal
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• v.17 no.1
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• pp.87-92
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• 2001

In this paper, we have the extended result of Bunce's theorem. And we show that if T is an irreducible p-hyponormal operator such that T*T-TT* is compact, then ${\sigma}_{ap}(T)={\sigma}_e(T)$ and ${\sigma}_p({\phi}(T))={\sigma}_e({\phi}(T))$.

• Bulletin of the Korean Mathematical Society
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• v.41 no.2
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• pp.337-345
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• 2004

In allelic model X = ($\chi_1\chi$_2ㆍㆍㆍ, \chi_d$), M_f(t) = f(p(t)) - ${{\int^t}_0}\;Lf(p(t))ds$ is a P-martingale for diffusion operator L under the certain conditions. In this note, we can show existence and uniqueness of solution for stochastic differential equation and martingale problem associated with mean vector. Also, we examine that if the operator related to this martingale problem is connected with Markov processes under certain circumstance, then this operator must satisfy the maximum principle.

• Bulletin of the Korean Mathematical Society
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• v.57 no.3
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• pp.535-545
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• 2020

Let A be an m × n matrix over nonnegative integers. The isolation number of A is the maximum number of isolated entries in A. We investigate linear operators that preserve the isolation number of matrices over nonnegative integers. We obtain that T is a linear operator that strongly preserve isolation number k for 1 ≤ k ≤ min{m, n} if and only if T is a (P, Q)-operator, that is, for fixed permutation matrices P and Q, T(A) = P AQ or, m = n and T(A) = P A^tQ for any m × n matrix A, where A^t is the transpose of A.

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1117-1127
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• 2019

Let ${\mathcal{L}}_2=(-{\Delta})^2+V^2$ be the $Schr{\ddot{o}}dinger$ type operator, where nonnegative potential V belongs to the reverse $H{\ddot{o}}lder$ class $RH_s$, s > n/2. In this paper, we consider the operator $T_{{\alpha},{\beta}}=V^{2{\alpha}}{\mathcal{L}}^{-{\beta}}_2$ and its conjugate $T^*_{{\alpha},{\beta}}$, where $0<{\alpha}{\leq}{\beta}{\leq}1$. We establish the $(L^p,\;L^q)$-boundedness of operator $T_{{\alpha},{\beta}}$ and $T^*_{{\alpha},{\beta}}$, respectively, we also show that $T_{{\alpha},{\beta}}$ is bounded from Hardy type space $H^1_{L_2}({\mathbb{R}}^n)$ into $L^{p_2}({\mathbb{R}}^n)$ and $T^*_{{\alpha},{\beta}}$ is bounded from $L^{p_1}({\mathbb{R}}^n)$ into BMO type space $BMO_{{\mathcal{L}}1}({\mathbb{R}}^n)$, where $p_1={\frac{n}{4({\beta}-{\alpha})}}$, $p_2={\frac{n}{n-4({\beta}-{\alpha})}}$.

• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.25-34
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• 2007

An operator $T{\in}L(H)$ is said to be p-paranormal if $$\parallel{\mid}T\mid^pU{\mid}T\mid^px{\parallel}x\parallel\geq\parallel{\mid}T\mid^px\parallel^2$$ for all $x{\in}H$ and p > 0, where $T=U{\mid}T\mid$ is the polar decomposition of T. It is easy that every 1-paranormal operator is paranormal, and every p-paranormal operator is paranormal for 0 < p < 1. In this note, we discuss some properties for p-paranormal operators.

• Journal of the Korean Mathematical Society
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• v.42 no.3
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• pp.529-541
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• 2005

An operator T belonging to the algebra B(H) of bounded linear transformations on a Hilbert H into itself is said to be posinormal if there exists a positive operator $P{\in}B(H)$ such that $TT^*\;=\;T^*PT$. A posinormal operator T is said to be conditionally totally posinormal (resp., totally posinormal), shortened to $T{\in}CTP(resp.,\;T{\in}TP)$, if to each complex number, $\lambda$ there corresponds a positive operator $P_\lambda$ such that $|(T-{\lambda}I)^{\ast}|^{2}\;=\;|P_{\lambda}^{\frac{1}{2}}(T-{\lambda}I)|^{2}$ (resp., if there exists a positive operator P such that $|(T-{\lambda}I)^{\ast}|^{2}\;=\;|P^{\frac{1}{2}}(T-{\lambda}I)|^{2}\;for\;all\;\lambda)$. This paper proves Weyl's theorem type results for TP and CTP operators. If $A\;{\in}\;TP$, if $B^*\;{\in}\;CTP$ is isoloid and if $d_{AB}\;{\in}\;B(B(H))$ denotes either of the elementary operators $\delta_{AB}(X)\;=\;AX\;-\;XB\;and\;\Delta_{AB}(X)\;=\;AXB\;-\;X$, then it is proved that $d_{AB}$ satisfies Weyl's theorem and $d^{\ast}_{AB}\;satisfies\;\alpha-Weyl's$ theorem.

• Journal of applied mathematics & informatics
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• v.27 no.3_4
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• pp.757-762
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• 2009

In allelic model $X\;=\;(x_1,\;x_2,\;{\cdots},\;x_d)$, $$M_f(t)\;=\;f(p(t))\;-\;{\int}^t_0\;Lf(p(t))ds$$ is a P-martingale for diffusion operator L under the certain conditions. We can also obtain a new diffusion operator $L^*$ for diffusion coefficient and we prove that unique solution for $L^*$-martingale problem exists. In this note, we define new symmetric preserving transformation. Uniqueness for martingale problem and symmetric property will be proved.