Linear preservers of polynomial numerical hulls of matrices

Abstract

Let Mn be the algebra of all n × n complex matrices, 1 ≤ k ≤ n − 1 be an integer, and φ : Mn −→ Mn be a linear operator. In this paper, it is shown that φ preserves the polynomial numerical hull of order k if and only if there exists a unitary matrix U ∈ Mn such that either φ(A)=U∗AU forallA∈Mn,orφ(A)=U∗AtU forallA∈Mn.