• Communications of the Korean Mathematical Society
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• v.12 no.4
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• pp.895-904
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• 1997

Let B(z) be a power series with operator coefficients where multiplication by B(z), T, is a contractive and everywhere defined transforamtion in the square summable power series. Then there is a Julia operator U for T such that $$ U = (T D)(\tilde{D}^* L) \in B(H \oplus D, K \oplus \tilde{D}), $$ where D is the state space of a conjugate canonical linear system with transfer function B(z).

• The Pure and Applied Mathematics
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• v.12 no.1
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• pp.75-91
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• 2005

The main objective of this paper is to introduce the gradation of neigh-bourhoodness in L-fuzzy topology and to introduce the fuzziness in the concept of convergence of L-fuzzy nets.

• Journal of applied mathematics & informatics
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• v.36 no.3_4
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• pp.237-244
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• 2018

Let ${\mathcal{H}}$ be an infinite dimensional separable Hilbert space with a fixed orthonormal base $\{e_1,e_2,{\cdots}\}$. Let L be the subspace lattice generated by the subspaces $\{[e_1],[e_1,e_2],[e_1,e_2,e_3],{\cdots}\}$ and let $Alg{\mathcal{L}}$ be the algebra of bounded operators which leave invariant all projections in ${\mathcal{L}}$. Let p and q be natural numbers (p < q). Let ${\mathcal{A}}$ be a linear manifold in $Alg{\mathcal{L}}$ such that $T_{(p,q)}=0$ for all T in ${\mathcal{A}}$. If ${\mathcal{A}}$ is a Lie ideal, then $T_{(p,p)}=T_{(p+1,p+1)}={\cdots}=T_{(q,q)}$ and $T_{(i,j)}=0$, $p{\eqslantless}i{\eqslantless}q$ and i < $j{\eqslantless}q$ for all T in ${\mathcal{A}}$.

• Journal of the Korean Mathematical Society
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• v.44 no.3
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• pp.627-645
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• 2007

We investigate the eigenvalues of the semi-circulant preconditioned matrix for the finite difference scheme corresponding to the second-order elliptic operator with the variable coefficients given by $L_vu\;:=-{\Delta}u+a(x,\;y)u_x+b(x,\;y)u_y+d(x,\;y)u$, where a and b are continuously differentiable functions and d is a positive bounded function. The semi-circulant preconditioning operator $L_cu$ is constructed by using the leading term of $L_vu$ plus the constant reaction term such that $L_cu\;:=-{\Delta}u+d_cu$. Using the field of values arguments, we show that the eigenvalues of the preconditioned matrix are clustered at some number. Some numerical evidences are also provided.

• 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 $K(E,\;f)\;=\;{\parallel}{\sum{\array}{n\\i=1}}\;E_{i}Y\;f_{i}{\parallel}/{\parallel}{\sum{\array}{n\\i=1}}\;E_{i}Xf_{i}{\parallel}$.

• Korean Journal of Mathematics
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• v.16 no.4
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• pp.583-587
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• 2008

Let $\mathfrak{L(H)}$ denote the algebra of bounded linear operators on a separable infinite dimensional complex Hilbert space $\mathfrak{H}$. We say that $T{\in}\mathfrak{L(H)}$ is a quasi-class A operator if $$T^*{\mid}T^2{\mid}T{{\geq}}T^*{\mid}T{\mid}^2T$$. In this paper we prove that if A and B are quasi-class A operators, and $B^*$ is invertible, then for a Hilbert-Schmidt operator X $$AX=XB\;implies\;A^*X=XB^*$$.

• Kyungpook Mathematical Journal
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• v.48 no.2
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• pp.281-286
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• 2008

In this note we prove that if A and $B^*$ are subnormal operators and is a bounded linear operator such that AX - XB is a Hilbert-Schmidt operator, then f(A)X - Xf(B) is also a Hilbert-Schmidt operator and $${\parallel}f(A)X\;-\;Xf(B){\parallel}_2\;\leq\;L{\parallel}AX\;-\;XB{\parallel}_2$$, for f belonging to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and $X\;{\in}\;\cal{L}(\cal{H})$ is such that SX - XT belongs to a norm ideal (J, ${\parallel}\;{\cdot}\;{\parallel}_J$) and prove that f(S)X - Xf(T) $\in$ J and ${\parallel}f(S)X\;-\;Xf(T){\parallel}_J\;\leq\;C{\parallel}SX\;-\;XT{\parallel}_J$, for f in a certain class of functions.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1209-1219
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• 2018

We provide a direct proof of the following theorem of Kalton, Hollenbeck, and Verbitsky [7]: let H be the Hilbert transform and let a, b be real constants. Then for 1 < p < ${\infty}$ the norm of the operator aI + bH from $L^p(\mathbb{R})$ to $L^p(\mathbb{R})$ is equal to $$\({\max_{x{\in}{\mathbb{R}}}}{\frac{{\mid}ax-b+(bx+a){\tan}{\frac{\pi}{2p}}{\mid}^p+{\mid}ax-b-(bx+a){\tan}{\frac{\pi}{2p}}{\mid}^p}{{\mid}x+{\tan}{\frac{\pi}{2p}}{\mid}^p+{\mid}x-{\tan}{\frac{\pi}{2p}}{\mid}^p}}\)^{\frac{1}{p}}$$. Our proof avoids passing through the analogous result for the conjugate function on the circle, as in [7], and is given directly on the line. We also provide new approximate extremals for aI + bH in the case p > 2.

• Kyungpook Mathematical Journal
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• v.55 no.4
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• pp.909-931
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• 2015

This paper presents some new paranormed sequence spaces $X(r,s,t,p;{\Delta})$ where $X{\in}\{l_{\infty}(p),c(p),c_0(p),l(p)\}$ defined by using generalized means and difference operator. It is shown that these are complete linear metric spaces under suitable paranorms. Furthermore, the ${\alpha}$-, ${\beta}$-, ${\gamma}$-duals of these sequence spaces are computed and also obtained necessary and sufficient conditions for some matrix transformations from $X(r,s,t,p;{\Delta})$ to X. Finally, it is proved that the sequence space $l(r,s,t,p;{\Delta})$ is rotund when $p_n$ > 1 for all n and has the Kadec-Klee property.

• Kyungpook Mathematical Journal
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• v.15 no.2
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• pp.191-199
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• 1975