Multiplicative trace and spectrum preservers on stochastic matrices
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915516976726016 |
|---|---|
| author | Tsai, Ming-Cheng Huang, Huajun |
| author_facet | Tsai, Ming-Cheng Huang, Huajun |
| contents | We characterize maps $ϕ_i: \mathcal{S} \to \mathcal{S}$, $i=1, \ldots, m$ and $m\ge 1$, that have the multiplicative spectrum or trace preserving property:
\begin{eqnarray*}
\textrm{spec} (ϕ_1(A_1)\cdots ϕ_m(A_m)) &=& \textrm{spec} (A_1\cdots A_m),\quad\text{or}\quad
\textrm{tr} (ϕ_1(A_1)\cdots ϕ_m(A_m)) &=& \textrm{tr} (A_1\cdots A_m),
\end{eqnarray*}
where $\mathcal{S}$ is the set of $n\times n$ doubly stochastic, row stochastic, or column stochastic matrices, or the space spanned by one of these sets.
Linearity is assumed when $m=1$. We show that every stochastic matrix contains a real doubly stochastic component that carries the spectral information.
In consequence, the multiplicative spectrum or trace preservers on these sets $ \mathcal{S} $ are linked to the
corresponding preservers on the space of doubly stochastic matrices.
Moreover, when $m\ge 3$, multiplicative trace preservers always coincide with multiplicative spectrum preservers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22743 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplicative trace and spectrum preservers on stochastic matrices Tsai, Ming-Cheng Huang, Huajun Functional Analysis Operator Algebras Probability Primary 15A86, Secondary 47B49, 15A15, 15A18 We characterize maps $ϕ_i: \mathcal{S} \to \mathcal{S}$, $i=1, \ldots, m$ and $m\ge 1$, that have the multiplicative spectrum or trace preserving property: \begin{eqnarray*} \textrm{spec} (ϕ_1(A_1)\cdots ϕ_m(A_m)) &=& \textrm{spec} (A_1\cdots A_m),\quad\text{or}\quad \textrm{tr} (ϕ_1(A_1)\cdots ϕ_m(A_m)) &=& \textrm{tr} (A_1\cdots A_m), \end{eqnarray*} where $\mathcal{S}$ is the set of $n\times n$ doubly stochastic, row stochastic, or column stochastic matrices, or the space spanned by one of these sets. Linearity is assumed when $m=1$. We show that every stochastic matrix contains a real doubly stochastic component that carries the spectral information. In consequence, the multiplicative spectrum or trace preservers on these sets $ \mathcal{S} $ are linked to the corresponding preservers on the space of doubly stochastic matrices. Moreover, when $m\ge 3$, multiplicative trace preservers always coincide with multiplicative spectrum preservers. |
| title | Multiplicative trace and spectrum preservers on stochastic matrices |
| topic | Functional Analysis Operator Algebras Probability Primary 15A86, Secondary 47B49, 15A15, 15A18 |
| url | https://arxiv.org/abs/2509.22743 |