• Communications of the Korean Mathematical Society
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• v.39 no.3
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• pp.717-729
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• 2024

Let 𝓑(H) be the algebra of all bounded linear operators on a Hilbert space H with dim(H) > 2. Let 𝒢𝒫(H) be the subset of 𝓑(H) of all generalized projection operators. In this paper, we give a complete characterization of surjective maps 𝚽 : 𝓑(H) → 𝓑(H) satisfying A-𝛌B ∈ 𝒢𝒫(H) ⇔ 𝚽(A) - 𝛌𝚽(B) ∈ 𝒢𝒫(H) for any A, B ∈ 𝓑(H) and 𝛌 ∈ ℂ.

• Honam Mathematical Journal
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• v.23 no.1
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• pp.51-57
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• 2001

Let $\mathcal{B}(H)$ and $\mathcal{B}(K)$ denote the algebras of all bounded linear operators on Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, respectively. We show that if ${\phi}:\mathcal{B}(H){\rightarrow}\mathcal{B}(K)$ is an additive mapping satisfying ${\phi}({\mid}A{\mid}^2)={\mid}{\phi}(A){\mid}^2$ for every $A{\in}\mathcal{B}(H)$, then there exists a mapping ${\psi}$ defined by ${\psi}(A)={\phi}(I){\phi}(A)$, ${\forall}A{\in}\mathcal{B}(H)$ such that ${\psi}$ is the sum of $two^*$-homomorphisms one of which C-linear and the othere C-antilinear. We will also study some conditions implying the injective and rank-preserving of ${\psi}$.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.541-547
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• 2008

Suppose R is an idempotence-diagonalizable ring. Let n and m be two arbitrary positive integers with $n\;{\geq}\;3$. We denote by $M_n(R)$ the ring of all $n{\times}n$ matrices over R. Let ($J_n(R)$) be the additive subgroup of $M_n(R)$ generated additively by all idempotent matrices. Let ($D=J_n(R)$) or $M_n(R)$. In this paper, by using an additive idem potence-preserving result obtained by Coo (see [4]), I characterize (i) the additive preservers of tripotence from D to $M_m(R)$ when 2 and 3 are units of R; (ii) the additive preservers of inverses (respectively, Drazin inverses, group inverses, {1}-inverses, {2}-inverses, {1, 2}-inverses) from $M_n(R)$ to $M_n(R)$ when 2 and 3 are units of R.

• Journal of the Korean Mathematical Society
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• v.54 no.2
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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 $A{\in}\mathcal{A}$.

• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.671-683
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• 2005

The set of all $m\;{\times}\;n$ matrices with entries in $\mathbb{Z}_+$ is de­noted by $\mathbb{M}{m{\times}n}(\mathbb{Z}_+)$. We say that a linear operator T on $\mathbb{M}{m{\times}n}(\mathbb{Z}_+)$ is a (U, V)-operator if there exist invertible matrices $U\;{\in}\; \mathbb{M}{m{\times}n}(\mathbb{Z}_+)$ and $V\;{\in}\;\mathbb{M}{m{\times}n}(\mathbb{Z}_+)$ such that either T(X) = UXV for all X in $\mathbb{M}{m{\times}n}(\mathbb{Z}_+)$, or m = n and T(X) = $UX^{t}V$ for all X in $\mathbb{M}{m{\times}n}(\mathbb{Z}_+)$. In this paper we show that a linear operator T preserves the rank of matrices over the nonnegative integers if and only if T is a (U, V)­operator. We also obtain other characterizations of the linear operator that preserves rank of matrices over the nonnegative integers.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.831-838
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• 2020

Let H be a complex Hilbert space and T : H → H be a bounded linear operator. Then T is said to be norm attaining if there exists a unit vector x₀ ∈ H such that ║Tx₀║ = ║T║. If for any closed subspace M of H, the restriction T|M : M → H of T to M is norm attaining, then T is called an absolutely norm attaining operator or 𝓐𝓝-operator. In this note, we discuss linear maps on B(H), which preserve the class of absolutely norm attaining operators on H.

• Journal of applied mathematics & informatics
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• v.19 no.1_2
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• pp.253-260
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• 2005

Denote the set of n ${\times}$ n complex Hermitian matrices by Hn. A pair of n ${\times}$ n Hermitian matrices (A, B) is said to be rank-additive if rank (A+B) = rank A+rank B. We characterize the linear maps from Hn into itself that preserve the set of rank-additive pairs. As applications, the linear preservers of adjoint matrix on Hn and the Jordan homomorphisms of Hn are also given. The analogous problems on the skew Hermitian matrix space are considered.

• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.115-122
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• 2004

We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. Let $S_{n}(F)$ be the space of all $n\;\times\;n$ symmetry matrices over a field F with 2, $3\;\in\;F^{*}$, then T is an additive injective operator preserving rank-additivity on $S_{n}(F)$ if and only if there exists an invertible matrix $U\;\in\;M_n(F)$ and an injective field homomorphism $\phi$ of F to itself such that $T(X)\;=\;cUX{\phi}U^{T},\;\forallX\;=\;(x_{ij)\;\in\;S_n(F)$ where $c\;\in;F^{*},\;X^{\phi}\;=\;(\phi(x_{ij}))$. As applications, we determine the additive operators preserving minus-order on $S_{n}(F)$ over the field F.

• Korean Journal of Mathematics
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• v.20 no.3
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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.

• Kyungpook Mathematical Journal
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• v.46 no.4
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• pp.581-589
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• 2006

We characterize the linear operators which preserve the factor rank of integer matrices. That is, if $\mathcal{M}$ is the set of all $m{\times}n$ matrices with entries in the integers and min($m,n$) > 1, then a linear operator T on $\mathcal{M}$ preserves the factor rank of all matrices in $\mathcal{M}$ if and only if T has the form either T(X) = UXV for all $X{\in}\mathcal{M}$, or $m=n$ and T(X)=$UX^tV$ for all $X{\in}\mathcal{M}$, where U and V are suitable nonsingular integer matrices. Other characterizations of factor rank-preservers of integer matrices are also given.