K-positivity preserver of matrix polynomials

For any closed K⊆ Rⁿ K ⊆ R n , recently all K -positivity preserver have been characterized, i.e., all linear operators T:R[x₁, ,xₙ]→ R[x₁, ,xₙ] T : R [ x 1 , ⋯ , x n ] → R [ x 1 , ⋯ , x n ] such that Tp≥ 0 T p ≥ 0 on K for all p≥ 0 p ≥ 0 on K . An important extension of polynomials R[x₁, ,xₙ] R [ x 1 , ⋯ , x n ] with real coefficients are polynomials Rm× m[x₁, ,xₙ] R m × m [ x 1 , ⋯ , x n ] with matrix coefficients. Non-negativity on K for matrix polynomials with Hermitian coefficients Hermₘ Herm m is then p(x) 0 p ( x ) ⪰ 0 for all x∈ K x ∈ K . In the current work, we investigate linear operators T:Hermₘ[x₁, ,xₙ]→ Hermₘ[x₁, ,xₙ] T : <mml