• Korean Journal of Mathematics
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• v.27 no.3
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• pp.735-741
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• 2019

Let ${\mathcal{H}}$ be a complex Hilbert space and ${\mathcal{B}}({\mathcal{H}})$ the algebra of all bounded linear operators on ${\mathcal{H}}$. In this paper, we prove that if ${\varphi}:{\mathcal{B}}({\mathcal{H}}){\rightarrow}{\mathcal{B}}({\mathcal{H}})$ is a unital surjective bounded linear map, which preserves m- isometries m = 1, 2 in both directions, then there are unitary operators $U,V{\in}{\mathcal{B}}({\mathcal{H}})$ such that ${\varphi}(T)=UTV$ or ${\varphi}(T)=UT^{tr}V$ for all $T{\in}{\mathcal{B}}({\mathcal{H}})$, where $T^{tr}$ is the transpose of T with respect to an arbitrary but fixed orthonormal basis of ${\mathcal{H}}$.

• Bulletin of the Korean Mathematical Society
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• v.60 no.6
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• pp.1687-1695
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• 2023

In this paper, we study the Green ring r(𝔴⁰_n) of the weak Hopf algebra 𝔴⁰_n based on Taft Hopf algebra H_n(q). Let R(𝔴⁰_n) := r(𝔴⁰_n) ⊗_ℤ ℂ be the Green algebra corresponding to the Green ring r(𝔴⁰_n). We first determine all finite dimensional simple modules of the Green algebra R(𝔴⁰_n), which is based on the observations of the roots of the generating relations associated with the Green ring r(𝔴⁰_n). Then we show that the nilpotent elements in r(𝔴⁰_n) can be written as a sum of finite dimensional indecomposable projective 𝔴⁰_n-modules. The Jacobson radical J(r(𝔴⁰_n)) of r(𝔴⁰_n) is a principal ideal, and its rank equals n - 1. Furthermore, we classify all finite dimensional non-simple indecomposable R(𝔴⁰_n)-modules. It turns out that R(𝔴⁰_n) has n² - n + 2 simple modules of dimension 1, and n non-simple indecomposable modules of dimension 2.

• 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 applied mathematics & informatics
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• v.10 no.1_2
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• pp.227-233
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• 2002

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$\sub$i/=Y$\sub$i/, for i=1,2, ‥‥, R. In this article, we investigate Hilbert-Schmidt interpolation for operators in tridiagonal algebras.

• Honam Mathematical Journal
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• v.39 no.1
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• pp.93-100
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• 2017

Let $\mathcal{H}$ be an infinite dimensional separable Hilbert space with a fixed orthonormal base $\{e_1,e_2,{\cdots}\}$. Let $\mathcal{L}$ 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($p{\leqslant}q$). Let $\mathcal{B}_{p,q}=\{T{\in}Alg\mathcal{L}{\mid}T_{(p,q)}=0\}$. Let $\mathcal{A}$ be a linear manifold in $Alg{\mathcal{L}}$ such that $\{0\}{\varsubsetneq}{\mathcal{A}}{\subset}{\mathcal{B}}_{p,q}$. If $\mathcal{A}$ is an ideal in $Alg{\mathcal{L}}$, then $T_{(i,j)}=0$, $p{\leqslant}i{\leqslant}q$ and $i{\leqslant}j{\leqslant}q$ for all T in $\mathcal{A}$.

• Bulletin of the Korean Mathematical Society
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• v.48 no.4
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• pp.673-688
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• 2011

Let $n{\geq}2$ be an even integer. We investigate that if an odd mapping f : X ${\rightarrow}$ Y satisfies the following equation $2_{n-2}C_{\frac{n}{2}-1}rf\(\sum\limits^n_{j=1}{\frac{x_j}{r}}\)\;+\;{\sum\limits_{i_k{\in}\{0,1\} \atop {{\sum}^n_{k=1}\;i_k={\frac{n}{2}}}}\;rf\(\sum\limits^n_{i=1}(-1)^{i_k}{\frac{x_i}{r}}\)=2_{n-2}C_{{\frac{n}{2}}-1}\sum\limits^n_{i=1}f(x_i),$ then f : X ${\rightarrow}$ Y is additive, where $r{\in}R$. We also prove the stability in normed group by using shadowing property and the Hyers-Ulam stability of the functional equation in Banach spaces and in Banach modules over unital C-algebras. As an application, we show that every almost linear bijection h : A ${\rightarrow}$ B of unital $C^*$-algebras A and B is a $C^*$-algebra isomorphism when $h(\frac{2^s}{r^s}uy)=h(\frac{2^s}{r^s}u)h(y)$ for all unitaries u ${\in}$ A, all y ${\in}$ A, and s = 0, 1, 2,....

• Communications of the Korean Mathematical Society
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• v.32 no.2
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• pp.261-266
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• 2017

H. R. Ebrahimi Vishki et al. conjectured in [1], that if every Jordan higher derivation on a trivial generalized matrix algebra $\mathcal{G}=(A,M,N,B)$ is a higher derivation, then either M = 0 or N = 0. In this note, we will give a class of counter examples.

• Korean Journal of Mathematics
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• v.20 no.2
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• pp.213-223
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• 2012

In this paper, we will prove the superstability of the following generalized Pexider type exponential equation $${f(x+y)}^m=g(x)h(y)$$, where $f,g,h\;:\;G{\rightarrow}\mathbb{R}$ are unknown mappings and $m$ is a fixed positive integer. Here G is an Abelian group (G, +), and $\mathbb{R}$ the set of real numbers. Also we will extend the obtained results to the Banach algebra. The obtained results are generalizations of P. G$\check{a}$vruta's result in 1994 and G. H. Kim's results in 2011.

• Bulletin of the Korean Mathematical Society
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• v.21 no.2
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• pp.107-113
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• 1984

Throughout the paper let A be a semifinite, infinite von Neumann algebra acting on a Hilbert space H, .alpha. an infinite cardinal. The main purpose of our work is to give several characterizations of a class of closed ideals in A, by introducing the notions of relative ranks of elements in A and the relative .alpha.-topology on H. The relative .alphi.-topology is an analogue to the .alpha.-topology that we have defined in ([7], [8]). The present work is regarded as an extension of [7], [8] and motivated by works of M. Breuer ([1], [2]), V. Kaftal ([5], [6]) and M.G. Sonis [9].

• 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.