• Honam Mathematical Journal
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• v.32 no.2
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• pp.255-260
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• 2010

Given vectors x and y in a separable complex Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that Ax = y. In this article, we investigate compact interpolation problems for vectors in a tridiagonal algebra. We show the following : Let Alg$\mathcal{L}$ be a tridiagonal algebra on a separable complex Hilbert space $\mathcal{H}$ and let x = $(x_i)$ and y = $(y_i)$ be vectors in H. Then the following are equivalent: (1) There exists a compact operator A = $(a_{ij})$ in Alg$\mathcal{L}$ such that Ax = y. (2) There is a sequence ${{\alpha}_n}$ in $\mathbb{C}$ such that ${{\alpha}_n}$ converges to zero and for all k ${\in}$ $\mathbb{N}$, $y_1 = {\alpha}_1x_1 + {\alpha}_2x_2$ $y_{2k} = {\alpha}_{4k-1}x_{2k}$ $y_{2k+1}={\alpha}_{4k}x_{2k}+{\alpha}_{4k+1}x_{2k+1}+{\alpha}_{4k+2}+x_{2k+2}$.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.597-602
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• 1997

Let H be a separable complex Hilbert space and L(H) be the *-algebra of all bounded linear operators on H. For $T \in L(H)$, we construct a pair of semi-positive definite operators $$ $\mid$T$\mid$_r = (T^*T)^{\frac{1}{2}} and $\mid$T$\mid$_l = (TT^*)^{\frac{1}{2}}. $$ An operator T is called a semi-hyponormal operator if $$ Q_T = $\mid$T$\mid$_r - $\mid$T$\mid$_l \geq 0. $$ In this paper, by using a technique introduced by Berberian [1], we show that the approximate point spectrum $\sigma_{ap}(T)$ of a semi-hyponomal operator T is empty.

• Journal of the Korean Mathematical Society
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• v.42 no.3
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• pp.405-434
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• 2005

We establish appropriate $L^p$ estimates for a class of maximal operators $S_{\Omega}^{(\gamma)}$ on the product space $R^n\;\times\;R^m\;when\;\Omega$ lacks regularity and $1\;\le\;\gamma\;\le\;2.\;Also,\;when\;\gamma\;=\;2$, we prove the $L^p\;(2\;{\le}\;P\;<\;\infty)\;boundedness\;of\;S_{\Omega}^{(\gamma)}\;whenever\;\Omega$ is a function in a certain block space $B_q^{(0,0)}(S^{n-1}\;\times\;S^{m-1})$ (for some q > 1). Moreover, we show that the condition $\Omega\;{\in}\;B_q^{(0,0)}(S^{n-1}\;\times\;S^{m-1})$ is nearly optimal in the sense that the operator $S_{\Omega}^{(2)}$ may fail to be bounded on $L^2$ if the condition $\Omega\;{\in}\;B_q^{(0,0)}(S^{n-1}\;\times\;S^{m-1})$ is replaced by the weaker conditions $\Omega\;{\in}\;B_q^{(0,\varepsilon)}(S^{n-1}\;\times\;S^{m-1})\;for\;any\;-1\;<\;\varepsilon\;<\;0.$

• Honam Mathematical Journal
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• v.28 no.1
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• pp.135-140
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• 2006

Given vectors x and y in a separable Hilbert space H, an interpolating operator is a bounded operator A such that Ax = y. In this article, we investigate self-adjoint interpolation problems for vectors in a tridiagonal algebra: Let AlgL be a tridiagonal algebra on a separable complex Hilbert space H and let $x=(x_i)$ and $y=(y_i)$ be vectors in H.Then the following are equivalent: (1) There exists a self-adjoint operator $A=(a_ij)$ in AlgL such that Ax = y. (2) There is a bounded real sequence {$a_n$} such that $y_i=a_ix_i$ for $i{\in}N$.

• Honam Mathematical Journal
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• v.36 no.1
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• pp.29-32
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• 2014

Given operators X and Y acting on a separable Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that AX = Y. In this article, we investigate self-adjoint interpolation problems for operators in a tridiagonal algebra : Let $\mathcal{L}$ be a subspace lattice acting on a separable complex Hilbert space $\mathcal{H}$ and let X = ($x_{ij}$) and Y = ($y_{ij}$) be operators acting on $\mathcal{H}$. Then the following are equivalent: (1) There exists a self-adjoint operator A = ($a_{ij}$) in $Alg{\mathcal{L}}$ such that AX = Y. (2) There is a bounded real sequence {${\alpha}_n$} such that $y_{ij}={\alpha}_ix_{ij}$ for $i,j{\in}\mathbb{N}$.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1715-1723
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• 2016

We study the characteristic Jacobi operator ${\ell}={\bar{R}({\cdot},{\xi}){\xi}$ (along the Reeb flow ${\xi}$) on the unit tangent sphere bundle $T_1M$ over a Riemannian manifold ($M^n$, g). We prove that if ${\ell}$ is pseudo-parallel, i.e., ${\bar{R}{\cdot}{\ell}=L{\mathcal{Q}}({\bar{g}},{\ell})$, by a non-positive function L, then M is locally flat. Moreover, when L is a constant and $n{\neq}16$, M is of constant curvature 0 or 1.

• Journal of the Korean Mathematical Society
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• v.38 no.1
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• pp.163-176
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• 2001

We examine Orlicz-type integral inequalities for operators and obtain as a corollary a characterization of such inequalities for the Hardy-Littlewood maximal operator extending the well-known L(sup)p-norm inequalities.

• Honam Mathematical Journal
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• v.34 no.3
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• pp.409-422
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• 2012

Let $\mathcal{L}$ be a subspace lattice on a Hilbert space $\mathcal{H}$. And let X and Y be operators acting on a Hilbert space $\mathcal{H}$. Let $sp(x)=\{{\alpha}x\;:\;{\alpha}{\in}\mathcal{C}\}$ $x{\in}\mathcal{H}$. Assume that $\mathcal{H}=\overline{range\;X}{\oplus}sp(h)$ for some $h{\in}\mathcal{H}$ and < $h$, $E^{\bot}Xf$ >= 0 for each $f{\in}\mathcal{H}$ and $E{\in}\mathcal{L}$. Then there exists an operator A in Alg$\mathcal{L}$ such that AX = Y if and only if $sup\{\frac{{\parallel}E^{\bot}Yf{\parallel}}{{\parallel}E^{\bot}Yf{\parallel}}\;:\;f{\in}H,\;E{\in}\mathcal{L}\}$ = K < ${\infty}$. Moreover, if the necessary condition holds, then we may choose an operator A such that AX = Y and ${\parallel}||A{\parallel}=K$.

• Korean Journal of Mathematics
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• v.6 no.2
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• pp.291-298
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• 1998

The $L^p-L^q$ mapping properties of convolution operators with measures supported on curves in $\mathbb{R}^3$ have been studied by many authors. Oberlin provided examples of nondegenerate compact space curves whose arclength measures enjoy $L^p$-improving properties. This was later extended by Pan who showed that such properties hold for all nondegenerate compact space curves. In this paper, we will prove that the operator norm of the convolution operator with the arclength measure supported on a nondegenerate compact space curve depends only on certain quantities of the underlying curve.

• Korean Journal of Mathematics
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• v.14 no.2
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• pp.137-160
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• 2006

We describe the weight functions for which Hardy's inequality of nonincreasing functions is satisfied. Further we characterize the pairs of weight functions $(w,v)$ for which the Laplace transform $\mathcal{L}f(x)={\int}^{\infty}_0e^{-xy}f(y)dy$, with monotone function $f$, is bounded from the weighted Lebesgue space $L^p(w)$ to $L^q(v)$.