On linear preservers of permanental rank

Let Mat_n(F) denote the set of square n×n matrices over a field F of characteristic different from two. The permanental rank prk(A) of a matrix A∈Mat_n(F) is the size of the maximal square submatrix in A with nonzero permanent. By Λ^k and Λ^≤k we denote the subsets of matrices A∈Mat_n(F) with prk(A)=k and prk(A)≤k, respectively. In this paper for each 1≤k≤n−1 we obtain a complete characterization of linear maps T:Mat_n(F)→Mat_n(F) satisfying T(Λ^≤k)=Λ^≤k or bijective linear maps satisfying T(Λ^≤k)⊆Λ^≤k. Moreover, we show that if F is an infinite field, then Λ^k is Zariski dense in Λ^≤k and apply this to describe such bijective linear maps satisfying T(Λ^k)⊆Λ^k.