Schur Multiplicative Maps on Matrices

Abstract

The structure of Schur multiplicative maps on matrices over a field is studied. The result is then used to characterize Schur multiplicative maps f satisfying f (S) ⊆ S for different subsets S of matrices including the set of rank k matrices, the set of singular matrices, and the set of invertible matrices. Characterizations are also obtained for maps on matrices such that ( f (A)) = (A) for various functions including the rank function, the determinant function, and the elementary symmetric functions of the eigenvalues. These results include analogs of the theorems of Frobenius and Dieudonné on linear maps preserving the determinant functions and linear maps preserving the set of singular matrices, respectively.2000 Mathematics subject classification: 15A03, 15A15.

Schur null preserving maps

“…Clearly, the map a → a ρ preserves the Schur product. Note that the same result has previously been obtained by Clark, Li and Rastogi in [3] with a different approach. Before moving into the infinite-dimensional matrices, we are interested to see if we could specify the permutations on the entries of the matrices with respect to our preservers.…”

“…Since the Schur product gives rise to a different algebraic structure, it is natural to seek for a characterisation of Schur multiplicative maps and Schur null preserving maps. While Schur multiplicative maps on matrices with some special properties such as T (S) ⊆ S or f (T (a)) = f (a) for a given function f on matrices were studied in [8,3,13], we are interested in a general characterisation of Schur multiplicative maps and maps preserving Schur zero product. In particular, we obtain a complete characterisation of contractive and completely contractive Schur multiplicative maps.…”

The Schur multiplicative and harmonic convexities of the complete symmetric function

“…Recently, the Schur multiplicative convexity was investigated in [13], [14], [21], [28], but no one has ever researched the Schur harmonic convexity. The aim of this article is to discuss the Schur multiplicative convexity and Schur harmonic convexity for the complete symmetric function F n (x, r) and the function ϕ n (x, r).…”

Schur ideals and homomorphisms of the semidefinite cone

“…These are special cases of Schur multiplicative maps (which preserve the Schur products) studied in [3].…”