• Communications of the Korean Mathematical Society
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• v.10 no.1
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• pp.235-249
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• 1995

Let K be a bounded linear operator from Hilbert space $H_1$ into Hilbert space $H_2$. When numerically solving the first kind equation Kf = g, one usually picks n orthonormal functions $\phi_1, \phi_2,...,\phi_n$ in $H_1$ which depend on the numerical method and on the problem, see Varah [12] for more details. Then one findes the unique minimum norm element $f_M \in M$ that satisfies $\Vert K f_M - g \Vert = inf {\Vert K f - g \Vert : f \in M}$, where M is the linear span of $\phi_1, \phi_2,...,\phi_n$. Such a solution $f_M \in M$ is called the M-solution of K f = g. Some methods for finding the M-solution of K f = g were proposed by Banks [2] and Marti [9,10]. See [5,6,8] for convergence results comparing the M-solution of K f = g with $f_0$, the least squares solution of minimum norm (LSSMN) of K f = g.

• Communications of the Korean Mathematical Society
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• v.32 no.3
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• pp.653-659
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• 2017

Let $p{\geq}1$ be a real number. A tuple $T=(T_1,{\ldots},T_n)$ of commuting bounded linear operators on a Banach space X is called an ${\ell}^p$-spherical isometry if ${\sum_{i=1}^{n}}{\parallel}T_ix{\parallel}^p={\parallel}x{\parallel}^p$ for all $x{\in}X$. The tuple T is called a toral isometry if each Ti is an isometry. By a result of Ansari, Hedayatian, Khani-Robati and Moradi, for every $n{\geq}1$, there is a supercyclic ${\ell}^2$-spherical isometric n-tuple on ${\mathbb{C}}^n$ but there is no supercyclic ${\ell}^2$-spherical isometry on an infinite-dimensional Hilbert space. In this article, we investigate the supercyclicity of ${\ell}^p$-spherical isometries and toral isometries on Banach spaces. Also, we introduce the notion of semicommutative tuples and we show that the Banach spaces ${\ell}^p$ ($1{\leq}p$ < ${\infty}$) support supercyclic ${\ell}^p$-spherical isometric semi-commutative tuples. As a result, all separable infinite-dimensional complex Hilbert spaces support supercyclic spherical isometric semi-commutative tuples.

• 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.33 no.3_4
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• pp.417-424
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• 2015

In this paper the following is proved: Let L be a subspace lattice on a Hilbert space H and X and Y be operators acting on a Hilbert space H. If XE = EX for each E ${\in}$ L, then there exists an operator A in AlgL such that AX = Y if and only if sup $\left{\frac{\parallel{XEf}\parallel}{\parallel{YEf}\parallel}\;:\;f{\in}H,\;E{\in}L\right}$ = K < $\infty$ and YE=EYE. Let x and y be non-zero vectors in H. Let P_x be the orthogonal pro-jection on sp(x). If EP_x = P_xE for each E $\in$ L, then the following are equivalent. (1) There exists an operator A in AlgL such that Ax = y. (2) < f, Ey > y =< f, Ey > Ey for each E ${\in}$ L and f ${\in}$ H.

• Honam Mathematical Journal
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• v.26 no.4
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• pp.379-389
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• 2004

Given operators X and Y acting on a Hilbert space ${\mathcal{H}}$, 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 $i=1,2,{\cdots},n$. In this article, we obtained the following : Let ${\mathcal{H}}$ be a Hilbert space and let ${\mathcal{L}}$ be a commutative subspace lattice on ${\mathcal{H}}$. Let X and Y be operators acting on ${\mathcal{H}}$. Then the following statements are equivalent. (1) There exists an operator A in $Alg{\mathcal{L}}$ such that AX = Y, A is positive and every E in ${\mathcal{L}}$ reduces A. (2) sup ${\frac{{\parallel}{\sum}^n_{i=1}\;E_iY\;f_i{\parallel}}{{\parallel}{\sum}^n_{i=1}\;E_iX\;f_i{\parallel}}}:n{\in}{\mathbb{N}},\;E_i{\in}{\mathcal{L}}$ and $f_i{\in}{\mathcal{H}}<{\infty}$ and <${\sum}^n_{i=1}\;E_iY\;f_i$, ${\sum}^n_{i=1}\;E_iX\;f_i>\;{\geq}0$, $n{\in}{\mathbb{N}}$, $E_i{\in}{\mathcal{L}}$ and $f_i{\in}H$.

• 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.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 >.

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.431-436
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• 2003

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 article, we obtained the following : $Let\;x\;=\;\{x_i\}\;and\;y=\{y_\}$ be two vectors in a separable complex Hilbert space H such that $x_i\;\neq\;0$ for all $i\;=\;1,\;2;\cdots$. Let L be a commutative subspace lattice on H. Then the following statements are equivalent. (1) $sup\;\{\frac{\$\mid${\sum_{k=1}}^l\;\alpha_{\kappa}E_{\kappa}y\$\mid$}{\$\mid${\sum_{k=1}}^l\;\alpha_{\kappa}E_{\kappa}x\$\mid$}\;:\;l\;\in\;\mathbb{N},\;\alpha_{\kappa}\;\in\;\mathbb{C}\;and\;E_{\kappa}\;\in\;L\}\;<\;\infty\;and\;$\mid$y_n\$\mid$x_n$\mid$^{-1}\;=\;1\;for\;all\;n\;=\;1,\;2,\;\cdots$. (2) There exists an operator A in AlgL such that Ax = y, A is a unitary operator and every E in L reduces, A, where AlgL is a tridiagonal algebra.

• Journal of KIISE:Databases
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• v.33 no.1
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• pp.32-41
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• 2006

In data-centric networks, various data items, such as temperature, humidity, etc. are sensed and stored in sensor nodes. As these attributes are mostly scalar values and inter-related, multi-dimensional range queries are useful. To process multi-dimensional range queries efficiently in data-centric storage, data addressing is essential. The Previous work focused on efficient query processing without considering overall network lifetime. To prolong network lifetime and support multi-dimensional range queries, we propose a dynamic data distribution method for multi-dimensional data, where data space is divided into equal-sized regions and linearized by using Hilbert space filling curve.

• Honam Mathematical Journal
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• v.23 no.1
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• pp.51-57
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• 2001

Let $\mathcal{B}(H)$ and $\mathcal{B}(K)$ denote the algebras of all bounded linear operators on Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, respectively. We show that if ${\phi}:\mathcal{B}(H){\rightarrow}\mathcal{B}(K)$ is an additive mapping satisfying ${\phi}({\mid}A{\mid}^2)={\mid}{\phi}(A){\mid}^2$ for every $A{\in}\mathcal{B}(H)$, then there exists a mapping ${\psi}$ defined by ${\psi}(A)={\phi}(I){\phi}(A)$, ${\forall}A{\in}\mathcal{B}(H)$ such that ${\psi}$ is the sum of $two^*$-homomorphisms one of which C-linear and the othere C-antilinear. We will also study some conditions implying the injective and rank-preserving of ${\psi}$.