Similarity preserving linear maps on upper triangular matrix algebras (Q819776)

Let \({\mathcal T}_n\) be the algebra of \(n\)-by-\(n\) upper-triangular complex matrices. Matrices \(A,B\in{\mathcal T}_n\) are similar if \(B=SAS^{-1}\) for some invertible \(S\in{\mathcal T}_n\). If \(S\) is allowed to come from \({\mathcal M}_n\) then \(A,B\) are called similar in \({\mathcal M}_n\). In parallel, a subspace \({\mathcal U}\subseteq{\mathcal T}_n\) is similarity invariant if \(S {\mathcal U}S^{-1}\subseteq {\mathcal U}\) for all upper-triangular, invertible \(S\). Moreover, it is similarity invariant in \({\mathcal M}_n\) if \(S {\mathcal U}S^{-1}\cap {\mathcal T}_n\subseteq {\mathcal U}\) for all invertible matrices \(S\in{\mathcal M}_n\). The authors determine all linear subspaces in \({\mathcal T}_n\) that are similarity invariant. They also determine all linear subspaces in \({\mathcal T}_n\) that are similarity invariant in \({\mathcal M}_n\). The two results are then used to classify linear bijections on upper-triangular matrices that (i) preserve similarity, and (ii) preserve similarity in \({\mathcal M}_n\).