On linear preservers of permanental rank

Let ${\mathrm{M}\mathrm{a}\mathrm{t}}_n(\mathbb{F})$ denote the set of square $n\times n$ matrices over a field $\mathbb{F}$ of characteristic different from two. The permanental rank $\mathrm{p}\mathrm{r}\mathrm{k}(A)$ of a matrix $A\in {\mathrm{M}\mathrm{a}\mathrm{t}}_n(\mathbb{F})$ is the size of the maximal square submatrix in $A$ with nonzero permanent. By ${\mathrm{\Lambda }}^k$ and ${\mathrm{\Lambda }}^{\le k}$ we denote the subsets of matrices $A\in {\mathrm{M}\mathrm{a}\mathrm{t}}_n(\mathbb{F})$ with $\mathrm{p}\mathrm{r}\mathrm{k}(A)=k$ and $\mathrm{p}\mathrm{r}\mathrm{k}(A)\le k$, respectively. In this paper for each $1\le k\le n-1$ we obtain a complete characterization of linear maps $T:{\mathrm{M}\mathrm{a}\mathrm{t}}_n(\mathbb{F})\to {\mathrm{M}\mathrm{a}\mathrm{t}}_n(\mathbb{F})$ satisfying $T({\mathrm{\Lambda }}^{\le k})={\mathrm{\Lambda }}^{\le k}$ or bijective linear maps satisfying $T({\mathrm{\Lambda }}^{\le k})\subseteq {\mathrm{\Lambda }}^{\le k}$. Moreover, we show that if $\mathbb{F}$ is an infinite field, then ${\mathrm{\Lambda }}^k$ is Zariski dense in ${\mathrm{\Lambda }}^{\le k}$ and apply this to describe such bijective linear maps satisfying $T({\mathrm{\Lambda }}^k)\subseteq {\mathrm{\Lambda }}^k$.