• Journal of applied mathematics & informatics
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• 제19권1_2호
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• pp.447-452
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• 2005

Given operators X and Y on a Hilbert space 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. Let L be a subspace lattice acting on a separable complex Hilbert space H and Alg L be a tridiagonal algebra. Let X = $(x_{ij})\;and\;Y\;=\;(y_{ij})$ be operators acting on H. Then the following are equivalent: (1) There exists a compact operator A = $(x_{ij})$ in AlgL such that AX = Y. (2) There is a sequence {$\alpha_n$} in $\mathbb{C}$ such that {$\alpha_n$} converges to zero and $$y_1\;_j=\alpha_1x_1\;_j+\alpha_2x_2\;_j\;y_{2k}\;_j=\alpha_{4k-1}x_{2k\;j}\;y_{2k+1\;j}=\alpha_{4k}x_{2k\;j}+\alpha_{4k+1}x_{2k+1\;j}+\alpha_{4k+2}x_{2k+2\;j\;for\;all\;k\;\epsilon\;\mathbb{N}$$.

• Korean Journal of Mathematics
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• 제6권2호
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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.

• 대한수학회보
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• 제33권3호
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• pp.445-453
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• 1996

A closed subspace J of a Banach space X is called an M-ideal in X if the annihilator $J^\perp$ of J is an L-summand of $X^*$. That is, there exists a closed subspace J' of $X^*$ such that $X^* = J^\perp \oplus J'$ and $\left\$\mid$ p + q \right\$\mid$ = \left\$\mid$ p \right\$\mid$ + \left\$\mid$ q \right\$\mid$$ wherever $p \in J^\perp and q \in J'$.

• 대한수학회보
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• 제49권5호
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• pp.899-910
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• 2012

Let $\mathcal{H}$ be a complex separable infinite dimensional Hilbert space. In this paper, a necessary and sufficient condition is given for an operator T on $\mathcal{H}$ to satisfy that $f(T)$ obeys generalized Weyl's theorem for each function $f$ analytic on some neighborhood of ${\sigma}(T)$. Also we investigate the stability of generalized Weyl's theorem under (small) compact perturbations.

• 대한수학회논문집
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• 제26권3호
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• pp.455-462
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• 2011

An analytic self-map ${\phi}$ of the open unit disk $\mathbb{D}$ in the complex plane and an analytic map ${\psi}$ on $\mathbb{D}$ induce the so-called weighted composition operator $C_{{\phi},{\psi}}$: $H(\mathbb{D})\;{\rightarrow}\;H(\mathbb{D})$, $f{\mapsto} \;{\psi}\;(f\;o\;{\phi})$, where H($\mathbb{D}$) denotes the set of all analytic functions on $\mathbb{D}$. We study when such an operator acting between different weighted Bergman spaces is bounded, compact and Schatten class.

• Korean Journal of Mathematics
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• 제16권4호
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• pp.545-551
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• 2008

We investigate the multiple solutions for a parabolic boundary value problem with jumping nonlinearity crossing no eigenvalues. We show the existence of the unique solution of the parabolic problem with Dirichlet boundary condition and periodic condition when jumping nonlinearity does not cross eigenvalues of the Laplace operator $-{\Delta}$. We prove this result by investigating the Lipschitz constant of the inverse compact operator of $D_t-{\Delta}$ and applying the contraction mapping principle.

• 대한수학회지
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• 제44권2호
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• pp.261-273
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• 2007

In this paper, we introduce some new properties of a topological space which are respectively generalizations of $Fr\'{e}chet$-Urysohn property. We show that countably AP property is a sufficient condition for a space being countable tightness, sequential, weakly first countable and symmetrizable, to be ACP, $Fr\'{e}chet-Urysohn$, first countable and semimetrizable, respectively. We also prove that countable compactness is a sufficient condition for a countably AP space to be countably $Fr\'{e}chet-Urysohn$. We then show that a countably compact space satisfying one of the properties mentioned here is sequentially compact. And we show that a countably compact and countably AP space is maximal countably compact if and only if it is $Fr\'{e}chet-Urysohn$. We finally obtain a sufficient condition for the ACP closure operator $[{\cdot}]_{ACP}$ to be a Kuratowski topological closure operator and related results.

• East Asian mathematical journal
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• 제14권2호
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• pp.311-316
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• 1998

We study some spectral properties of the Laplace-Beltrami operator on compact Riemannian manifolds.

• 대한수학회지
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• 제38권6호
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• pp.1157-1166
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• 2001

This paper answers some old questions about approximate similarity and raises new ones. We provide positive evidence and a technique for finding negative evidence on the question of whether approximate similarity is the equivalence relation generated by approximate equivalence and similarity.

• Korean Journal of Mathematics
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• 제30권2호
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• pp.305-313
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• 2022

In this paper we consider a modified version of Smirnov operator and obtain some Bernstein-type inequalities preserved by this operator. In particular, we prove some results which in turn provide the compact generalizations of some well-known inequalities for polynomials.