10 problems

Let $m,n\geq2$, and let a real $m\times n$ matrix be semipositive if it maps some vector in $\mathbb{R}^n$ to a vector whose entries are all positive. An into linear preserver of s…

Let $\mathcal{S}_n$ be the real symmetric $n\times n$ matrices, and let $\operatorname{COP}_n$ denote the cone of copositive matrices. An into copositivity preserver is a linear ma…

Let $m=ngeq4$, and let $\mathcal{S}_n$ denote the real symmetric $n\times n$ matrices. A linear operator $\mathcal{L}:\mathcal{S}_n\to\mathcal{S}_n$ preserves cp-rank if it maps ev…

Scalarity conjecture. Based on the authors' investigations, one has

Rank-$k$ operator Schmidt preserver conjecture. The following conditions are equivalent:

Maribel et al.'s conjecture. For $n\geq 3$, every linear preserver of the L-spectrum must have the form

Dorsey et al.'s conjecture. If $L:M_{m,n}\to M_{m,n}$ is invertible and is an into preserver of $S(\mathbb{R}^n_+,\mathbb{R}^m_+)$, then there exist matrices $X$ and $Y$ such that

Let $cmathbb{K}$ be a field, let $cmathrm{M}_n(cmathbb{K})$ be the vector space of $n\times n$ matrices over $cmathbb{K}$, and let $cmathrm{DG}_n(cmathbb{K})$ be the subset of diag…

Let $H$ be an infinite-dimensional complex Hilbert space, and let $\mathcal{B}(H)$ denote the algebra of all bounded linear operators on $H$. A unital surjective linear map on…

Let ${\mathbb F}$ be a field, let ${\cal M}_n({\mathbb F})$ be the algebra of $n\times n$ matrices over ${\mathbb F}$, and let…