相关论文: Linear k-power preservers and trace of power-produ…

Let $M_n(\mathbb{F})$ denote the algebra of $n \times n$ matrices over an algebraically closed field $\mathbb{F}$ of characteristic different from $2$. For $n \ge 2$, we classify all maps $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$…

Let $n_1,\ldots,n_k $ be integers larger than or equal to 2. We characterize linear maps $\phi: M_{n_1\cdots n_k}\rightarrow M_{n_1\cdots n_k}$ such that $${\mathrm rank}\,(\phi(A_1\otimes \cdots \otimes…

Let $m,n\ge 2$ be integers. Denote by $M_n$ the set of $n\times n$ complex matrices. Let $\|\cdot\|_{(p,k)}$ be the $(p,k)$ norm on $M_{mn}$ with $1\leq k\leq mn$ and $2<p<\infty$. We show that a linear map $\phi:M_{mn}\rightarrow M_{mn}$…

For any closed $K\subseteq\mathbb{R}^n$, in [P.\ J.\ di\,Dio, K.\ Schm\"udgen: $K$-Positivity Preserver and their Generators, SIAM J.\ Appl.\ Algebra Geom.\ 9 (2025), 794--824] all $K$-positivity preserver have been characterized, i.e., all…

Let $m,n\ge 2$ be positive integers, $M_m$ the set of $m\times m$ complex matrices and $M_n$ the set of $n\times n$ complex matrices. Regard $M_{mn}$ as the tensor space $M_m\otimes M_n$. Suppose $|\cdot|$ is the Ky Fan $k$-norm with $1 \le…

We prove the linearity and injectivity of two maps $\phi_1$ and $\phi_2$ on certain subsets of $M_n$ that satisfy $\operatorname{tr}(\phi_1(A)\phi_2(B))=\operatorname{tr}(AB)$. We apply it to characterize maps $\phi_i:\mathcal{S}\to…

Let $\mathcal H$ be a complex Hilbert space and $\mathcal F_s (\mathcal H)$ the real vector space of all self-adjoint finite rank bounded operators on $\mathcal H$. We generalize the famous Wigner's theorem by characterizing linear maps on…

This paper solves the following problem about Hermitian matrices related to the theory of $2$-structures:\emph{ }Let $n$ be a positive integer and $k$ be an integer with $k\in \{3,\ldots,n-3\}$. Characterize the Hermitian matrices $A$ such…

For a positive integer $n$, let $M_n$ be the set of $n\times n$ complex matrices. Suppose $\|\cdot\|$ is the Ky Fan $k$-norm with $1 \le k \le mn$ or the Schatten $p$-norm with $1 \le p \le \infty$ ($p\ne 2$) on $M_{mn}$, where $m,n\ge 2$…

Let ${\rm Mat}_n(\mathbb{F})$ denote the set of square $n\times n$ matrices over a field $\mathbb{F}$ of characteristic different from two. The permanental rank ${\rm prk}\,(A)$ of a matrix $A \in{\rm Mat}_{n}(\mathbb{F})$ is the size of…

Let ${\mathcal M}_2(\mathbb F)$ be the algebra of 2$\times$2 matrices over the real or complex field $\mathbb F$. For a given positive integer $k\geq 1$, the $k$-commutator of $A$ and $B$ is defined by $[A,B]_k=[[A,B]_{k-1},B]$ with…

Let $\phi: A\to A$ be a (not necessarily linear, additive or continuous) map of a standard operator algebra. Suppose for any $a,b\in A$ there is an algebra automorphism $\theta_{a,b}$ of $ A$ such that \begin{align*} \phi(a)\phi(b) =…

For positive integers $1 \leq k \leq n$ let $M_n$ be the algebra of all $n \times n$ complex matrices and $M_n^{\le k}$ its subset consisting of all matrices of rank at most $k$. We first show that whenever $k>\frac{n}{2}$, any continuous…

We study linear functions on the space of $n \times n$ matrices over a field which preserve or strongly preserve each of Green's equivalence relations ($\mathcal{L}$, $\mathcal{R}$, $\mathcal{H}$ and $\mathcal{J}$) and the corresponding…

We study the problem of characterizing linear preserver subgroups of algebraic varieties, with a particular emphasis on secant varieties and other varieties of tensors. We introduce a number of techniques built on different geometric…

We characterise all Jordan triple product homomorphisms, that is, mappings $\Phi$ satisfying $$ \Phi(ABA) = \Phi(A)\Phi(B)\Phi(A) $$ on the set of all Hermitian $2 \times 2$ complex matrices.

We use classical results from the theory of linear preserver problems to characterize operators that send the set of pure states with Schmidt rank no greater than k back into itself, extending known results characterizing operators that…

We study $K$-positivity preservers with given closed $K\subseteq\mathbb{R}^n$, i.e., linear maps $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ such that $T\mathrm{Pos}(K)\subseteq\mathrm{Pos}(K)$ holds, and their generators…

We characterise all Jordan triple product homomorphisms, that is, mappings $\Phi$ satisfying $$ \Phi(ABA) = \Phi(A)\Phi(B)\Phi(A) $$ from the set of all Hermitian $n \times n$ complex matrices to the field of complex numbers. Further we…

Let $\psi : G\to GL(V)$ and $\varphi :G \to GL (W)$ be representations of finite group $G$. A linear map $T: V\to W$ is called a morphism from $\psi$ to $\varphi$ if it satisfys $T\psi_g= \varphi_g T$ for each $g\in G$ and let…