Generalization of numerical range of polynomial operator matrices

Abstract

Suppose that Q(λ)=λ^m A_m+λ^(m-1) A_(m-1)+⋯+A_0 is a polynomial matrix operator where A_i∈M_n (C) for i=0,1,…,m, are n×n complex matrix and let λ be a complex variable. For an n×n Hermitian matrix S, we define the V-numerical range of polynomial matrix of Q(λ) asV_S (Q(λ))={ λ∈C; ⟨Q(λ)x,x⟩=0,"for some" x∈C^n,⟨x,x⟩_S≠0 }, where ⟨x,y⟩_S=y^* Sx. In this paper we study V_S (Q(λ)) and our emphasis is on the geometrical properties of V_S (Q(λ)). We consider the location of V_S (Q(λ)) in the complex plane and a theorem concerning the boundary of V_S (Q(λ)) is also obtained. Possible generalazations of our results including their extensions to bounded linerar operators on an infinite dimensional Hilbert space are described.