Some multiplicative preservers on $BH)$

随机矩阵上的乘法谱保持子与迹保持子

Multiplicative trace and spectrum preservers on stochastic matrices

本文研究随机矩阵集合上的乘法谱与迹保持映射，通过随机矩阵所含的实双随机谱分量，将其归结为双随机矩阵空间上的保持子，并证明 $m\ge3$ 时两类保持子等价。

Some linear preserver problems on B(H) concerning rank and corank

As a continuation of the work on linear maps between operator algebras which preserve certain subsets of operators with finite rank, or corank, here we consider the problem inbetween, that is, we treat the question of preserving operators with infinite rank and infinite corank. Since, as it turns out, in this generality our preservers cannot be written in a nice form what we have got used to when dealing with linear preserver problems, hence we restrict our attention to certain important classes of operators like idempotents, or projections, or partial isometries. We conclude the paper with a …

Linear k-power preservers and trace of power-product preservers

Let $V$ be the set of $n\times n$ complex or real general matrices, Hermitian matrices, symmetric matrices, positive definite (resp. semi-definite) matrices, diagonal matrices, or upper triangular matrices. Fix $k\in \mathbb{Z}\setminus \{0, 1\}$. We characterize linear maps $ψ:V\to V$ that satisfy $ψ(A^k)=ψ(A)^k$ on an open neighborhood $S$ of $I_n$ in $V$. The $k$-power preservers are necessarily $k$-potent preservers, and the case $k=2$ corresponds to Jordan homomorphisms. Applying the results, we characterize maps $ϕ,ψ:V\to V$ that satisfy "$ \operatorname{tr}(ϕ(A)ψ(B)^k)=\operatorname{tr}…

A linear preserver problem on maps which are triple derivable at orthogonal pairs

A linear mapping $T$ on a JB$^*$-triple is called triple derivable at orthogonal pairs if for every $a,b,c\in E$ with $a\perp b$ we have $$0 = \{T(a), b,c\} + \{a,T(b),c\}+\{a,b,T(c)\}.$$ We prove that for each bounded linear mapping $T$ on a JB$^*$-algebra $A$ the following assertions are equivalent: $(a)$ $T$ is triple derivable at zero; $(b)$ $T$ is triple derivable at orthogonal elements; $(c)$ There exists a Jordan $^*$-derivation $D:A\to A^{**}$, a central element $ξ\in A^{**}_{sa},$ and an anti-symmetric element $η$ in the multiplier algebra of $A$, such that $$ T(a) = D(a) + ξ\circ a +…

New multiplicative perturbation bounds for the generalized polar decomposition

Some new Frobenius norm bounds of the unique solution to certain structured Sylvester equation are derived. Based on the derived norm upper bounds, new multiplicative perturbation bounds are provided both for subunitary polar factors and positive semi-definite polar factors. Some previous results are then improved.

Šemrl 保域定理的一个变体：针对奇异矩阵

A variant of Šemrl's preserver theorem for singular matrices

本文证明了一个 Šemrl 保域定理的变体：对于秩至多 k 的复矩阵集，在连续、单射、保持交换性且收缩谱的条件下，映射必为相似或转置共轭形式，并讨论了奇异矩阵情形及假设的必要性。