• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.635-647
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• 1995

Let $H$ denote a separable, infinite dimensional complex Hilbert space and let $L(H)$ denote the algebra of all bounded linear operators on $H$. A dual algebra is a subalgebra of $L(H)$ that contains the identity operator $1_H$ and is closed in the $weak^*$ operator topology on $L(H)$. For $T \in L(H)$, let $A_T$ denote the smallest subalgebra of $L(H)$ that contains T and $1_H$ and is closed in the $weak^*$ operator topology.

• Journal of the Korean Mathematical Society
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• v.34 no.1
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• pp.167-179
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• 1997

In the study of a manifold M, the exterior algebra $\Lambda^* M$ plays an important role. In fact, the de Rham cohomology theory gives many informations of a manifold. Another important object in the study of a manifold is its Clifford algebra (Cl(M), generated by the tangent space.

• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.311-319
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• 1995

Let $H$ be a separable, infinite dimensional, complex Hilbert space and let $L(H)$ be the algebra of all bounded linear operaors on $H$. A dual algebra is a subalgebra of $L(H)$ that contains the identity operator $I_H$ and is closed in the $weak^*$ topology on $L(H)$. Note that the ultraweak operator topology coincides with the $weak^*$ topology on $L(H)$ (see [5]).

• Honam Mathematical Journal
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• v.46 no.1
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• pp.82-100
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• 2024

The main objective of the study is to introduce ideals of Sheffer stroke Hilbert algebras by means of fuzzy points, and investigate some properties. The process of making (fuzzy) ideals and fuzzy deductive systems through the fuzzy points of Sheffer stroke Hilbert algebras is illustrated, and the (fuzzy) ideals and the fuzzy deductive systems are characterized. Certain sets are defined by virtue of a fuzzy set, and the conditions under which these sets can be ideals are revealed. The union and intersection of two fuzzy ideals are analyzed, and the relationships between aforementioned structures of Sheffer stroke Hilbert algebras are built.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1275-1283
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• 2010

If the Wiener-Hopf $C^*$-algebra W(G,M) for a discrete group G with a semigroup M has the uniqueness property, then the structure of it is to some extent independent of the choice of isometries on a Hilbert space. In this paper we show that if the Wiener-Hopf $C^*$-algebra W(G,M) of a partially ordered group G with the positive cone M has the uniqueness property, then (G,M) is weakly unperforated. We also prove that the Wiener-Hopf $C^*$-algebra W($\mathbb{Z}$, M) of subsemigroup generating the integer group $\mathbb{Z}$ is isomorphic to the Toeplitz algebra, but W($\mathbb{Z}$, M) does not have the uniqueness property except the case M = $\mathbb{N}$.

• Journal of the Korean Mathematical Society
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• v.42 no.3
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• pp.573-597
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• 2005

A continuous linear operator T, on the space of entire functions in d variables, is PDE-preserving for a given set $\mathbb{P}\;\subseteq\;\mathbb{C}|\xi_{1},\ldots,\xi_{d}|$ of polynomials if it maps every kernel-set ker P(D), $P\;{\in}\;\mathbb{P}$, invariantly. It is clear that the set $\mathbb{O}({\mathbb{P}})$ of PDE-preserving operators for $\mathbb{P}$ forms an algebra under composition. We study and link properties and structures on the operator side $\mathbb{O}({\mathbb{P}})$ versus the corresponding family $\mathbb{P}$ of polynomials. For our purposes, we introduce notions such as the PDE-preserving hull and basic sets for a given set $\mathbb{P}$ which, roughly, is the largest, respectively a minimal, collection of polynomials that generate all the PDE-preserving operators for $\mathbb{P}$. We also describe PDE-preserving operators via a kernel theorem. We apply Hilbert's Nullstellensatz.

• Honam Mathematical Journal
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• v.27 no.2
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• pp.243-250
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• 2005

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. We show the following: 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 an invertible operator $A=(a_{ij})$ in $Alg{\mathcal{L}}$ such that AX = Y. (2) There exist bounded sequences {${\alpha}_n$} and {${\beta}_n$} in ${\mathbb{C}}$ such that $${\alpha}_{2k-1}{\neq}0,\;{\beta}_{2k-1}=\frac{1}{{\alpha}_{2k-1}},\;{\beta}_{2k}=-\frac{{\alpha}_{2k}}{{\alpha}_{2k-1}{\alpha}_{2k+1}}$$ and $$y_{i1}={\alpha}_1x_{i1}+{\alpha}_2x_{i2}$$ $$y_{i\;2k}={\alpha}_{4k-1}x_{i\;2k}$$ $$y_{i\;2k+1}={\alpha}_{4k}x_{i\;2k}+{\alpha}_{4k+1}x_{i\;2k+1}+{\alpha}_{4k+2}x_{i\;2k+2}$$ for $$k{\in}N$$.

• Honam Mathematical Journal
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• v.33 no.1
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• pp.115-120
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• 2011

Given vectors x and y in a separable complex Hilbert space $\cal{H}$, an interpolating operator is a bounded operator A such that Ax = y. We show the following : Let Alg$\cal{L}$ 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 an invertible operator A = ($a_{kj}$) in Alg$\cal{L}$ such that Ax = y. (2) There exist bounded sequences $\{{\alpha}_n\}$ and $\{{{\beta}}_n\}$ in $\mathbb{C}$ such that for all $k\in\mathbb{N}$, ${\alpha}_{2k-1}\neq0,\;{\beta}_{2k-1}=\frac{1}{{\alpha}_{2k-1}},\;{\beta}_{2k}=\frac{\alpha_{2k}}{{\alpha}_{2k-1}\alpha_{2k+1}}$ and $$y_1={\alpha}_1x_1+{\alpha}_2x_2$$ $$y_{2k}={\alpha}_{4k-1}x_{2k}$$ $$y_{2k+1}={\alpha}_{4k}x_{2k}+{\alpha}_{4k+1}x_{2k+1}+{\alpha}_{4k+2}x_{2k+2}$$.

• Nonlinear Functional Analysis and Applications
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• v.27 no.2
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• pp.249-259
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• 2022

Let 𝓐 be a unital C^*-algebra. Then it follows that $\sum\limits_{i=1}^{n}(a_ia^*_i)^{\frac{1}{2}}{\leq}\sqrt{n}\(\sum\limits_{i=1}^{n}a_ia^*_i\)^{\frac{1}{2}}$, ∀n ∈ ℕ, ∀a₁, …, a_n ∈ 𝓐. By modifications of arguments of Botelho-Andrade, Casazza, Cheng, and Tran given in 2019, for certain n-tuple x = (a₁, …, a_n) ∈ 𝓐ⁿ, we give a method to compute a positive element c_x in the C^*-algebra 𝓐 such that the equality $$\sum\limits_{i=1}^{n}(a_ia^*_i)^{\frac{1}{2}}=c_x\sqrt{n}\(\sum\limits_{i=1}^{n}a_ia^*_i\)^{\frac{1}{2}}$$ holds. We give an application for the integral of Kasparov. We also derive a formula for the exact constant for the continuous ℓ₁ - ℓ₂ inequality.

• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.405-411
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• 2003

We prove that 25 different Artinian O-Sequences of codimension 3 type 3 with length 7 among all possible 334 cases are not level.