Matrix positivity preservers in fixed dimension. I

Matrix positivity preservers in fixed dimension

A celebrated 1942 result of Schoenberg characterizes all entry-wise functions which preserve positivity of matrices of any size. I will present a characterization of polynomials which preserve positivity when applied entry-wise on matrices of a fixed dimension. All put in historical context and motivated by recent demands of statistics of large data and optimization theory. A sketch of the proo...

A Note on Positivity Preservers

In this short follow-up of [3] we answer some questions posed there. In particular, we characterize two natural classes of linear operators acting on R[x1, x2, ..., xn] which (i) preserve the set of positive n-variate polynomials; (ii) either act diagonally in the monomial basis of R[x1, x2, ..., xn] or are translation-invariant.

Representation and Approximation of Positivity Preservers

We consider a closed set S ⊆ Rn and a linear operator Φ: R[X1, . . . , Xn]→ R[X1, . . . , Xn] that preserves nonnegative polynomials, in the following sense: if f ≥ 0 on S, then Φ(f) ≥ 0 on S as well. We show that each such operator is given by integration with respect to a measure taking nonnegative functions as its values. This can be seen as a generalization of Haviland’s Theorem, which conc...

On Nonlinear Preservers of Weak Matrix Majorization

Linear preservers of two-sided matrix majorization

For vectors X, <span style="fon...