• Journal of applied mathematics & informatics
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• 제40권5_6호
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• pp.1129-1135
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• 2022

The purpose of this research is to find an extension of the renowned Chernoff theorem on standard operator algebra. Infact, we prove the following result: Let H be a real (or complex) Banach space and 𝓛(H) be the algebra of bounded linear operators on H. Let 𝓐(H) ⊂ 𝓛(H) be a standard operator algebra. Suppose that D : 𝓐(H) → 𝓛(H) is a linear mapping satisfying the relation D(AⁿBⁿ) = D(Aⁿ)Bⁿ + AⁿD(Bⁿ) for all A, B ∈ 𝓐(H). Then D is a linear derivation on 𝓐(H). In particular, D is continuous. We also present the limitations on such identity by an example.

• 대한수학회보
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• 제44권2호
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• pp.271-279
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• 2007

Let A standard operator algebra which does not contain the identity operator, acting on a Hilbert space of dimension greater than one. If ${\Phi}$ is a bijective Lie map from A onto an arbitrary algebra, that is $${\phi}$$(AB-BA)=$${\phi}(A){\phi}(B)-{\phi}(B){\phi}(A)$$ for all A, B${\in}$A, then ${\phi}$ is additive. Also, if A contains the identity operator, then there exists a bijective Lie map of A which is not additive.

• 대한수학회지
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• 제54권2호
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• pp.563-574
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• 2017

Let $\mathcal{A}$ be a unital real standard Jordan operator algebra acting on a Hilbert space H of dimension at least 2. We show that every bijection ${\phi}$ on $\mathcal{A}$ satisfying ${\phi}(A^2{\circ}B)={\phi}(A)^2{\circ}{\phi}(B)$ is of the form ${\phi}={\varepsilon}{\psi}$ where ${\psi}$ is an automorphism on $\mathcal{A}$ and ${\varepsilon}{\in}\{-1,1\}$. As a consequence if $\mathcal{A}$ is the real algebra of all self-adjoint operators on a Hilbert space H, then there exists a unitary or conjugate unitary operator U on H such that ${\phi}(A)={\varepsilon}UAU^*$ for all $A{\in}\mathcal{A}$.

• 대한수학회보
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• 제47권4호
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• pp.787-792
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• 2010

Let A and B be some standard operator algebras on complex Banach spaces X and Y, respectively. We give the concrete forms of linear idempotence preserving maps $\Phi\;:\;A\;{\rightarrow}\;B$ on finite-rank operators.

• Korean Journal of Mathematics
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• 제20권3호
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• pp.285-291
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• 2012

Let $M_n$ be the algebra of all complex $n{\times}n$ matrices and ${\phi}:M_n{\rightarrow}M_n$ a surjective map (not necessarily additive or multiplicative) satisfying one of the following equations: $${\det}({\phi}(A){\phi}(B)+{\phi}(X))={\det}(AB+X),\;A,B,X{\in}M_n,\\{\sigma}({\phi}(A){\phi}(B)+{\phi}(X))={\sigma}(AB+X),\;A,B,X{\in}M_n$$. Then it is an automorphism, where ${\sigma}(A)$ is the spectrum of $A{\in}M_n$. We also show that if $\mathfrak{A}$ be a standard operator algebra, $\mathfrak{B}$ is a unital Banach algebra with trivial center and if ${\phi}:\mathfrak{A}{\rightarrow}\mathfrak{B}$ is a multiplicative surjection preserving spectrum, then ${\phi}$ is an algebra isomorphism.

• 대한수학회지
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• 제54권4호
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• pp.1063-1079
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• 2017

Two standard Bessel sequences in a Hilbert $C^*$-module are approximately duals if the distance (with respect to the norm) between the identity operator on the Hilbert $C^*$-module and the operator constructed by the composition of the synthesis and analysis operators of these Bessel sequences is strictly less than one. In this paper, we introduce (a, m)-approximate duality using the distance between the identity operator and the operator defined by multiplying the Bessel multiplier with symbol m by an element a in the center of the $C^*$-algebra. We show that approximate duals are special cases of (a, m)-approximate duals and we generalize some of the important results obtained for approximate duals to (a, m)-approximate duals. Especially we study perturbations of (a, m)-approximate duals and (a, m)-approximate duals of modular Riesz bases.