Some linear preserver problems on block triangular matrix algebras (Q1405024)

A given sequence of positive integers \(n_1,\dots,n_k\) with sum \(n\) determines a partitioning of \(n\times n\) matrices over a field \(F\) into \(n_i \times n_j\) blocks. The corresponding block upper-triangular matrices form a subalgebra \(J(n_1, \dots, n_k)\) of the total matrix algebra \(M_n\). A linear mapping \(\Psi:J(n_1, \dots, n_k)\to J(n_1, \dots, n_k)\) is said to be rank-\(r\) nonincreasing if rank \(\Psi(A)\leq r\) whenever rank \(A\leq r\). In theorem 3.5, the authors determine in explicit detail those rank-1 nonincreasing \(\Psi\) for which the image of the unit matrix \(I_n\) in non-singular. In a companion theorem 4.2, the authors determine, for \(n\geq 3\), those \(\Psi\) that commute with the adjugate mapping: \(adj(\Psi(A))= \Psi(adj A)\). There are other similar results, all surprisingly complicated in form.