• Journal of applied mathematics & informatics
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• v.41 no.2
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• pp.273-286
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• 2023

In this paper we show that if T ∈ L(X) and S ∈ L(X) is a Riesz operator commuting with T and X_S(F) ∈ Lat(S), where F = {0} or F ⊆ ℂ ⧵ {0} is closed then T|X_S(F) and T|X_T(F) + S|X_S(F) share the local spectral properties such as SVEP, Dunford's property (C), Bishop's property (𝛽), decomopsition property (𝛿) and decomposability. As a corollary, if T ∈ L(X) and Q ∈ L(X) is a quasinilpotent operator commuting with T then T is Riesz if and only if T + Q is Riesz. We also study some spectral properties of Riesz operators acting on Banach spaces. We show that if T, S ∈ L(X) such that TS = ST, and Y ∈ Lat(S) is a hyperinvarinat subspace of X for which 𝜎(S|Y ) = {0} then 𝜎_*(T|Y + S|Y ) = 𝜎_*(T|Y ) for 𝜎_* ∈ {𝜎, 𝜎_loc, 𝜎_sur, 𝜎_ap}. Finally, we show that if T ∈ L(X) and S ∈ L(Y ) on the Banach spaces X and Y and T is similar to S then T is Riesz if and only if S is Riesz.

• Korean Journal of Mathematics
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• v.22 no.1
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• pp.91-121
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• 2014

Let X and Y be vector spaces. It is shown that a mapping $f:X{\rightarrow}Y$ satisfies the functional equation ${\ddag}$ $$2df(\frac{x_1+{\sum}_{j=2}^{2d}(-1)^jx_j}{2d})-2df(\frac{x_1+{\sum}_{j=2}^{2d}(-1)^{j+1}x_j}{2d})=2\sum_{j=2}^{2d}(-1)^jf(x_j)$$ if and only if the mapping $f:X{\rightarrow}Y$ is additive, and prove the Cauchy-Rassias stability of the functional equation (${\ddag}$) in Banach modules over a unital $C^*$-algebra, and in Poisson Banach modules over a unital Poisson $C^*$-algebra. Let $\mathcal{A}$ and $\mathcal{B}$ be unital $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras. As an application, we show that every almost homomorphism $h:\mathcal{A}{\rightarrow}\mathcal{B}$ of $\mathcal{A}$ into $\mathcal{B}$ is a homomorphism when $h(d^nuy)=h(d^nu)h(y)$ or $h(d^nu{\circ}y)=h(d^nu){\circ}h(y)$ for all unitaries $u{\in}\mathcal{A}$, all $y{\in}\mathcal{A}$, and n = 0, 1, 2, ${\cdots}$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras, and of Lie $JC^*$-algebra derivations in Lie $JC^*$-algebras.

• Journal of the Korean Mathematical Society
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• v.47 no.2
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• pp.235-246
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• 2010

A Banach space operator A $\in$ B(X) is polaroid if the isolated points of the spectrum of A are poles of the resolvent of A; A is hereditarily polaroid, A $\in$ ($\mathcal{H}\mathcal{P}$), if every part of A is polaroid. Let $X^n\;=\;\oplus^n_{t=i}X_i$, where $X_i$ are Banach spaces, and let A denote the class of upper triangular operators A = $(A_{ij})_{1{\leq}i,j{\leq}n$, $A_{ij}\;{\in}\;B(X_j,X_i)$ and $A_{ij}$ = 0 for i > j. We prove that operators A $\in$ A such that $A_{ii}$ for all $1{\leq}i{\leq}n$, and $A^*$ have the single-valued extension property have spectral properties remarkably close to those of Jordan operators of order n and n-normal operators. Operators A $\in$ A such that $A_{ii}$ $\in$ ($\mathcal{H}\mathcal{P}$) for all $1{\leq}i{\leq}n$ are polaroid and have SVEP; hence they satisfy Weyl's theorem. Furthermore, A+R satisfies Browder's theorem for all upper triangular operators R, such that $\oplus^n_{i=1}R_{ii}$ is a Riesz operator, which commutes with A.

• Journal of the Korean Mathematical Society
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• v.51 no.2
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• pp.251-266
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• 2014

In this paper, we use Newton's method to approximate a locally unique solution of an equation in Banach spaces and introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton's method than before [1]-[13], in some interesting cases, provided that the Fr$\acute{e}$chet-derivative of the operator involved is p-H$\ddot{o}$lder continuous (p${\in}$(0, 1]). Numerical examples involving two boundary value problems are also provided.

• Bulletin of the Korean Mathematical Society
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• v.27 no.2
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• pp.189-196
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• 1990

In this paper, we try to generalize ultraproducts in the category of locally convex spaces. To do so, we introduce D-ultracolimits. It is known [7] that the topology on a non-trivial ultraproduct in the category T $V^{ec}$ of topological vector spaces and continuous linear maps is trivial. To generalize the category Ba $n_{1}$ of Banach spaces and linear contractions, we introduce the category L $C_{1}$ of vector spaces endowed with families of semi-norms closed underfinite joints and linear contractions (see Definition 1.1) and its subcategory, L $C_{2}$ determined by Hausdorff objects of L $C_{1}$. It is shown that L $C_{1}$ contains the category LC of locally convex spaces and continuous linear maps as a coreflective subcategory and that L $C_{2}$ contains the category Nor $m_{1}$ of normed linear spaces and linear contractions as a coreflective subcategory. Thus L $C_{1}$ is a suitable category for the study of locally convex spaces. In L $C_{2}$, we introduce $l_{\infty}$(I. $E_{i}$ ) for a family ( $E_{i}$ )$_{i.mem.I}$ of objects in L $C_{2}$ and then for an ultrafilter u on I. we have a closed subspace $N_{u}$ . Using this, we construct ultraproducts in L $C_{2}$. Using the relationship between Nor $m_{1}$ and L $C_{2}$ and that between Nor $m_{1}$ and Ba $n_{1}$, we show thatour ultraproducts in Nor $m_{1}$ and Ba $n_{1}$ are exactly those in the literatures. For the terminology, we refer to [6] for the category theory and to [8] for ultraproducts in Ba $n_{1}$..

• Korean Journal of Mathematics
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• v.19 no.1
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• pp.17-24
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• 2011

Let X, Y be vector spaces. It is shown that if an even mapping $f:X{\rightarrow}Y$ satisfies f(0) = 0, and $$2(_{2d-2}C_{d-1}-_{2d-2}C_d)f\({\sum_{j=1}^{2d}}x_j\)+{\sum_{{\iota}(j)=0,1,{{\small\sum}_{j=1}^{2d}}{\iota}(j)=d}}\;f\({\sum_{j=1}^{2d}}(-1)^{{\iota}(j)}x_j\)=2(_{2d-1}C_d+_{2d-2}C_{d-1}-_{2d-2}C_d){\sum_{j=1}^{2d}}f(x_j)$$ for all $x_1$, ${\cdots}$, $x_{2d}{\in}X$, then the even mapping $f:X{\rightarrow}Y$ is quadratic. Furthermore, we prove the Hyers-Ulam stability of the above functional equation in Banach spaces.

• Journal of the Chungcheong Mathematical Society
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• v.30 no.2
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• pp.239-249
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• 2017

In this paper, we approximate a fuzzy almost cubic function by a cubic function in a fuzzy sense. Indeed, we investigate solutions of the following cubic functional equation $$3f(kx+y)+3f(kx-y)-kf(x+2y)-2kf(x-y)-3k(2k^2-1)f(x)+6kf(y)=0$$. and prove the generalized Hyers-Ulam stability for it in fuzzy Banach spaces.

• Bulletin of the Korean Mathematical Society
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• v.31 no.2
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• pp.297-308
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• 1994

In [5], M. Cwikel and J. Peetre showed that some Banach couples are Calderon pairs and every K-monotone cones in that couples can be constructed by the Peetre K-method. With the help of theorem 4 in [3], we can extend the above result to the K-monotone spaces. In [4], M. Cwikel provided the proof of therem 4 in [3] and also mentioned that some theorems in [5] which were stated in terms of K-monotone cones can be extended to K-monotone spaces. But M. Cwikel and J.Peetre's results only cover the case when over bar A = over bar B. We extend this result to the case when over bar A.neq. over bar B. In this case, we need a condition stronger than that the K-monotone space.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.3
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• pp.397-408
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• 2015

In this paper, we solve the Hyers-Ulam stability problem for the following cubic type functional equation $$f(rx+sy)+f(rx-sy)=rs^2f(x+y)+rs^2f(x-y)+2r(r^2-s^2)f(x)$$in quasi-Banach space and non-Archimedean space, where $r={\neq}{\pm}1,0$ and s are real numbers.

• Journal of the Chungcheong Mathematical Society
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• v.17 no.1
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• pp.57-75
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• 2004

The aim of this paper is to solve the general solution of a quadratic functional equation f(x + 2y) + 2f(x - y) = f(x - 2y) + 2f(x + y) in the class of functions between real vector spaces and to obtain the generalized Hyers-Ulam stability problem for the equation.