• Honam Mathematical Journal
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• v.30 no.2
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• pp.329-334
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• 2008

Given operators X and Y acting on a Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that AX = Y. In this article, the following is proved: Let $\mathcal{L}$ be a subspace lattice on $\mathcal{H}$ and let X and Y be operators acting on a Hilbert space H. Let P be the projection onto the $\overline{rangeX}$. If PE = EP for each E ${\in}$ $\mathcal{L}$, then the following are equivalent: (1) sup ${{\frac{{\parallel}E^{\perp}Yf{\parallel}}{{\parallel}E^{\perp}Xf{\parallel}}}:f{\in}\mathcal{H},\;E{\in}\mathcal{L}}$ < ${\infty},\;\overline{rangeY}\;{\subset}\;\overline{rangeX}$, and there is a bounded operator T acting on $\mathcal{H}$ such that < Xf, Tg >=< Yf, Xg >, < Tf, Tg >=< Yf, Yg > for all f and gin $\mathcal{H}$ and $T^*h$ = 0 for h ${\in}\;{\overline{rangeX}}^{\perp}$. (2) There is a normal operator A in AlgL such that AX = Y and Ag = 0 for all g in range ${\overline{rangeX}}^{\perp}$.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.293-299
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• 2005

Let ${\cal{L}}$ be a commutative subspace lattice on a Hilbert space ${\cal{H}}$ and X and Y be operators on ${\cal{H}}$. Let $${\cal{M}}_X=\{{\sum}{\limits_{i=1}^n}E_{i}Xf_{i}:n{\in}{\mathbb{N}},f_{i}{\in}{\cal{H}}\;and\;E_{i}{\in}{\cal{L}}\}$$ and $${\cal{M}}_Y=\{{\sum}{\limits_{i=1}^n}E_{i}Yf_{i}:n{\in}{\mathbb{N}},f_{i}{\in}{\cal{H}}\;and\;E_{i}{\in}{\cal{L}}\}.$$ Then the following are equivalent. (i) There is an operator A in $Alg{\cal{L}}$ such that AX = Y, Ag = 0 for all g in ${\overline{{\cal{M}}_X}}^{\bot},A^*A=AA^*$ and every E in ${\cal{L}}$ reduces A. (ii) ${\sup}\;\{K(E, f)\;:\;n\;{\in}\;{\mathbb{N}},f_i\;{\in}\;{\cal{H}}\;and\;E_i\;{\in}\;{\cal{L}}\}\;<\;\infty,\;{\overline{{\cal{M}}_Y}}\;{\subset}\;{\overline{{\cal{M}}_X}}$and there is an operator T acting on ${\cal{H}}$ such that ${\langle}EX\;f,Tg{\rangle}={\langle}EY\;f,Xg{\rangle}$ and ${\langle}ET\;f,Tg{\rangle}={\langle}EY\;f,Yg{\rangle}$ for all f, g in ${\cal{H}}$ and E in ${\cal{L}}$, where $K(E,\;f)\;=\;{\parallel}{\sum{\array}{n\\i=1}}\;E_{i}Y\;f_{i}{\parallel}/{\parallel}{\sum{\array}{n\\i=1}}\;E_{i}Xf_{i}{\parallel}$.

• East Asian mathematical journal
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• v.37 no.1
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• pp.41-77
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• 2021

Let M be a semi-invariant submanifold of codimension 3 with almost contact metric structure (��, ξ, η, g) in a complex space form Mn+1(c), c ≠ 0. We denote by Rξ = R(·, ξ)ξ and A(i) be Jacobi operator with respect to the structure vector field ξ and be the second fundamental form in the direction of the unit normal C(i), respectively. Suppose that the third fundamental form t satisfies dt(X, Y ) = 2��g(��X, Y ) for certain scalar ��(≠ 2c)and any vector fields X and Y and at the same time Rξ is ��∇ξξ-parallel, then M is a Hopf hypersurface in Mn(c) provided that it satisfies RξA(1) = A(1)Rξ, RξA(2) = A(2)Rξ and ${\bar{r}}-2(n-1)c{\leq}0$, where ${\bar{r}}$ denotes the scalar curvature of M.

• Korean Journal of Mathematics
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• v.19 no.1
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• pp.1-16
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• 2011

In this paper, using differential operator, we have introduce new class of p-valent uniformly convex functions in the unit disc U = {z : |z| < 1} and obtain the coefficient bounds, extreme bounds and radius of starlikeness for the functions belonging to this generalized class. Furthermore, partial sums $f_k(z)$ of functions $f(z)$ in the class $S^*({\lambda},{\alpha},{\beta})$ are considered. The various results obtained in this paper are sharp.

• East Asian mathematical journal
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• v.38 no.1
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• pp.43-63
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• 2022

Let M be a semi-invariant submanifold of codimension 3 with almost contact metric structure (φ, ξ, η, g) in a complex space form M_n+1(c). We denote by R_ξ the structure Jacobi operator with respect to the structure vector field ξ and by ${\bar{r}}$ the scalar curvature of M. Suppose that R_ξ is φ∇_ξξ-parallel and at the same time the third fundamental form t satisfies dt(X, Y) = 2θg(φX, Y) for a scalar θ(≠ 2c) and any vector fields X and Y on M. In this paper, we prove that if it satisfies R_ξφ = φR_ξ, then M is a Hopf hypersurface of type (A) in M_n+1(c) provided that ${\bar{r}-2(n-1)c}$ ≤ 0.

• Journal of the Chungcheong Mathematical Society
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• v.13 no.1
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• pp.79-85
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• 2000

In this paper, we have an affirmative solution of G. Godini's open question ([3]): If a normed almost linear space Y is complete, is the normed almost linear space B(X, (Y,C)) complete?

• Journal of the Korean Mathematical Society
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• v.45 no.5
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• pp.1203-1220
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• 2008

Let V be an arbitrary system of weights on an open connected subset G of ${\mathbb{C}}^N(N{\geq}1)$ and let B (E) be the Banach algebra of all bounded linear operators on a Banach space E. Let $HV_b$ (G, E) and $HV_0$ (G, E) be the weighted locally convex spaces of vector-valued analytic functions. In this paper, we characterize self-analytic mappings ${\phi}:G{\rightarrow}G$ and operator-valued analytic mappings ${\Psi}:G{\rightarrow}B(E)$ which generate weighted composition operators and invertible weighted composition operators on the spaces $HV_b$ (G, E) and $HV_0$ (G, E) for different systems of weights V on G. Also, we obtained compact weighted composition operators on these spaces for some nice classes of weights.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.1
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• pp.81-95
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• 2007

We are concerned with the multiplicity of solutions of the nonlinear biharmonic equation with Dirichlet boundary condition, ${\Delta}^2u+c{\Delta}u=g(u)$, in ${\Omega}$, where $c{\in}R$ and ${\Delta}^2$ denotes the biharmonic operator. We show that there exists at least three solutions of the above problem under the suitable condition of g(u).

• The Pure and Applied Mathematics
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• v.23 no.1
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• pp.13-19
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• 2016

In this paper, we study some characterizations of unbounded domains. Among these, so-called G-domain is introduced by Cabre for the Aleksandrov-Bakelman-Pucci maximum principle of second order linear elliptic operator in a non-divergence form. This domain is generalized to wG-domain by Vitolo for the maximum principle of an unbounded domain, which contains G-domain. We study the properties of these domains and compare some other characterizations. We prove that sA-domain is wG-domain, but using the Cantor set, we are able to construct a example which is wG-domain but not sA-domain.

• The Journal of the Korea Contents Association
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• v.19 no.7
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• pp.186-191
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• 2019

Introduction: In aged people, stroke incidence is increased. But standardized experimental animal protocol study for the research of stroke therapy is rare. There is little report on the success rate of cerebral artery occlusion model using standardized Nylon thread length of precise thread end-size controlled. Method: In this study, the operator intended the occlusion of middle cerebral artery (MCA) using $0.18{\pm}0.02mm$ end 5-0 Nylon thread. Middle cerebral artery occlusion was induced for 60min under isoflurane anesthesia. After 60min, the operator removed the Nylon thread and reperfusion was induced for 23hrs. The mice was killed 23hrs after reperfusion and infarction area of brain was confirmed by 1.5% TTC (2,3,5-tryphenyl tetrazolium chloride) staining. Results: According to end size and insert length of Nylon thread, Middle cerebral artery occlusion (n=50), internal carotid artery occlusion (n= 14), distal middle cerebral artery occlusion (n= 36), anterior cerebral artery (n= 1) were induced. And no infarction (n= 50) was observed. Conclusion: According to weight of mice, the operator induced reversible cerebral artery occlusion model by different insert length (30.0~36.9g : 9.0mm, 37.0~40.0g : 9.5mm) of Nylon thread. Success of cerebral artery occlusion model was confirmed by checking infarction area using TTC staining. The success rate (66.9%, 101/151) of reversible cerebral artery occlusion model in the mouse and the operational conditions are shown.