Polynomial numerical hulls of matrix polynomials

Let M(n) be the algebra of all n x n complex matrices. Suppose Q(lambda) = A(m)lambda(m) + A(m-1)lambda(m-1) + center dot center dot center dot + A(1)lambda + A(0) is a matrix polynomial, where A(i) is an element of M(n) (i = 0, 1,..., m) and lambda is a complex variable. In this note, we introduce the polynomial numerical hull and the joint polynomial numerical hull of order k of Q(lambda). The emphasis is on the geometric properties of polynomial numerical hulls of matrix polynomials. Also, we study the relationships among spectra, numerical ranges and polynomial numerical hulls of order k of matrix polynomials.