Multiplicative preservers on semigroups of matrices
A ``linear preserver'' is a linear function of the space of all matrices (of given dimension over a given field) into itself which maps a prescribed subset \(\mathcal{P}\) into itself (for example, \(\mathcal{P}\) might be the set of all nonsingular matrices). Linear preservers have been widely studied and a typical theme is that for many interesting choices of \(\mathcal{P}\) they are necessarily of one of the forms: \(A\mapsto PAQ\) or \(A\mapsto PA^{T}Q\) for invertible \(P\) and \(Q\) [see, for example, \textit{C. K. Li} and \textit{S. Pierce}, Am. Math. Mon. 108, 591-605 (2001; Zbl 0991.15001)]. In the present paper the authors consider a related problem of ``multiplicative preservers''. Let \(\mathbf{S}\) be either a field or the semigroup of nonnegative reals or nonnegative rationals, and let \(\mathcal{R}\) be a subsemigroup of the semigroup \(M_{n}(\mathbf{S})\) of all \(n\times n\) matrices over \({\mathbf{S}}\). Consider a multiplicative map \(\phi\) of \(\mathcal{R}\) into itself. The authors prove that for certain subsets \(\mathcal{P\subseteq R}\), if \(\phi\) preserves \(\mathcal{P}\), then one of the following typically holds: (i) there exists an invertible \(S\) and a nonzero ``additive-multiplicative'' map \(f:\mathbf{S}\rightarrow\mathbf{S}\) such that \(\phi\) has one of the forms \(A\mapsto S^{-1}[f(a_{ij})]S\) or \(A\mapsto S^{-1}[f(\det A_{ij})S\) where \(A=[a_{ij}]\) and \(A_{ij}\) is the submatrix obtained by deleting the \(i\)th row and \(j\)th column (the precise definition of ``additive-multiplicative'' is too complicated to give here); (ii) there exists an invertible \(S\), an integer \(q\) with \(0\leq q\leq n\) and a multiplicative map \(\psi:M_{n}(\mathbf{S})\rightarrow M_{n-q}(\mathbf{S})\) such that \(\phi\) has the form \(A\mapsto S^{-1}(I_{q}\oplus\psi(A))S\); (iii) \(\phi\) is identically \(0\) on all matrices of rank \(\leq p\) where \(p\) depends on the subset preserved. For example, they prove such a theorem when \(\mathcal{P}\) is the set of rank-\(k\) matrices, the totally nonnegative matrices and the Hermitian matrices, and in each case give additional conditions which must hold. They also prove analogous theorems for multiplicative functions \(\phi\) which preserve functions \(g\) in the sense \(g(\phi(A))=g(A)\); for example, where \(g\) is the \(2\)-norm or the spectral radius.
scientific article; zbMATH DE number 1864740