• The Pure and Applied Mathematics
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• v.11 no.1
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• pp.29-36
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• 2004

Given vectors x and ｙ in a Hilbert space, an interpolating operator is a bounded operator T such that Tx=y. In this paper the following is proved: Let $\cal{L}$ be a subspace lattice on a Hilbert space $\cal{H}$. Let x and ｙ be vectors in $\cal{H}$ and let $P_x$, be the projection onto sp(x). If $P_xE=EP_x$ for each $ E \in \cal{L}$ then the following are equivalent. (1) There exists an operator A in Alg(equation omitted) such that Ax=y, Af = 0 for all f in ($sp(x)^\perp$) and $A=-A^\ast$. (2) (equation omitted)

• Korean Journal of Mathematics
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• v.15 no.2
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• pp.135-148
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• 2007

We prove the stability of generalized Jensen's equations in a Hilbert module over a unital $C^*$-algebra. This is applied to show the stability of a projection, a unitary operator, a self-adjoint operator, a normal operator, and an invertible operator in a Hilbert module over a unital $C^*$-algebra.

• Journal of applied mathematics & informatics
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• v.7 no.1
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• pp.61-82
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• 2000

In this paper two so-called regularized Green's functions are introduced to derive the optimal maximum norm error estimates for the unknown function and the adjoint vector-valued function for mixed finite element methods of Laplacian operator. One contribution of the paper is a demonstration of how the boundedness of $L^1$-norm estimate for the second Green's function ${\lambda}_2$ and the optimal maximum norm error estimate for the adjoint vector-valued function are proved. These results are seemed to be to be new in the literature of the mixed finite element methods.

• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.387-395
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• 2004

Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. An interpolating operator for n-operators satisfies the equation $AX_{i}\;=\;Y_{i}$, for i = 1,2,...,n. In this article, we showed the following: Let H be a Hilbert space and let L be a subspace lattice on H. Let X and Y be operators acting on H. Assume that range(X) is dense in H. Then the following statements are equivalent: (1) There exists an operator A in AlgL such that AX = Y, $A^{*}$ = A and every E in L reduces A. (2) sup ${\frac{$\mid$$\mid${\sum_{i=1}}^n\;E_iYf_i$\mid$$\mid$}{$\mid$$\mid${\sum_{i=1}}^n\;E_iXf_i$\mid$$\mid$}$:n{\epsilon}N,f_i{\epsilon}H\;and\;E_i{\epsilon}L}\;<\;{\infty}$ and = for all E in L and all f, g in H.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.503-510
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• 2004

Given vectors x and y in a Hilbert space, an interpolating operator is a bounded operator T such that Tx = y. An interpolating operator for n vectors satisfies the equation $Tx_i\;=\;y_i,\;for\;i\;=\;1,\;2,\;\cdots,\;n$. In this paper the following is proved: Let H be a Hilbert space and L be a commutative subspace lattice on H. Let H and y be vectors in H. Let $M_x\;=\;\{{\sum{n}{i=1}}\;{\alpha}_iE_ix\;:\;n\;{\in}\;N,\;{\alpha}_i\;{\in}\;{\mathbb{C}}\;and\;E_i\;{\in}\;L\}\;and\;M_y\;=\;\{{\sum{n}{i=1}}\;{\alpha}_iE_iy\;:\;n\;{\in}\;N,\;{\alpha}_i\;{\in}\;{\mathbb{C}}\;and\;E_i\;{\in}\;L\}. Then the following are equivalent. (1) There exists an operator A in AlgL such that Ax = y, Af = 0 for all f in ${\overline{M_x}}^{\bot}$, AE = EA for all $E\;{\in}\;L\;and\;A^{*}\;=\;A$. (2) $sup\;\{\frac{{\parallel}{{\Sigma}_{i=1}}^{n}\;{\alpha}_iE_iy{\parallel}}{{\parallel}{{\Sigma}_{i=1}}^{n}\;{\alpha}_iE_iy{\parallel}}\;:\;n\;{\in}\;N,\;{\alpha}_i\;{\in}\;{\mathbb{C}}\;and\;E_i\;{\in}\;L\}\;<\;{\infty},\;{\overline{M_u}}\;{\subset}{\overline{M_x}}$ and < Ex, y >=< Ey, x > for all E in L.

• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.981-986
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• 2011

Given vectors x and y in a Hilbert space $\cal{H}$, an interpolating operator is a bounded operator T such that Tx = y. An interpolating operator for n vectors satisfies the equations $Tx_i=y_i$, for i = 1, 2, ${\cdots}$, n. In this paper the following is proved : Let $\cal{L}$ be a subspace lattice on a Hilbert space $\cal{H}$. Let x and y be vectors in $\cal{H}$ and let $P_x$ be the projection onto sp(x). If $P_xE=EP_x$ for each $E{\in}\cal{L}$, then the following are equivalent. (1) There exists an operator A in Alg$\cal{L}$ such that Ax = y, Af = 0 for all f in $sp(x)^{\perp}$ and $A=A^*$. (2) sup $sup\;\{\frac{{\parallel}E^{\perp}y{\parallel}}{{\parallel}E^{\perp}x{\parallel}}\;:\;E\;{\in}\;{\cal{L}}\}$ < ${\infty}$, $y\;{\in}\;sp(x)$ and < x, y >=< y, x >.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1137-1147
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• 2018

In this paper, we discovered a sufficient condition ensuring the weighted composition operators $C_{{\omega}_1,{\varphi}_1},{\cdots},C_{{\omega}_N,{\varphi}_N}$ were disjoint supercyclic on $H({\Omega})$ endowed with the compact open topology. Besides, we provided a condition on inducing symbols to guarantee the disjoint supercyclicity of non-constant adjoint multipliers $M^*_{{\varphi}_1},M^*_{{\varphi}_2},{\cdots},M^*_{{\varphi}_N}$ on a Hilbert space ${\mathcal{H}}$.

• Korean Journal of Mathematics
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• v.24 no.1
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• pp.65-70
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• 2016

J. G. Stampfli proved that if a bounded linear operator T on a Hilbert space ${\mathfrak{H}}$ satisfies ($G_1$) property, then the Riesz projection $P_{\lambda}$ associated with ${\lambda}{\in}iso{\sigma}$(T) is self-adjoint and $P_{\lambda}{\mathfrak{H}}=(T-{\lambda})^{-1}(0)=(T^*-{\bar{\lambda}})^{-1}(0)$. In this note we show that Stampfli''s result is generalized to an nilpotent extension of an operator having ($G_1$) property.

• Korean Journal of Mathematics
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• v.27 no.4
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• pp.879-898
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• 2019

In this paper we prove some interesting inclusions concerning the numerical range of some operators and the numerical range of theirs ranges with a convex function. Further, we prove some inequalities for the numerical radius. These inclusions and inequalities are based on some classical convexity inequalities for non-negative real numbers and some operator inequalities.

• Korean Journal of Mathematics
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• v.29 no.3
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• pp.511-518
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• 2021

In this paper, a relationship between the extended spectrum of the Aluthge transform and the extended spectrum of the operator T is proved. Other relationships between two different operators and other results are also given.