The q-numerical range of matrix polynomials

Let P(lambda) = A(m)lambda (m) + A(m-1)lambda (m-1) +...+A(1)lambda +A(0) b e a matrix polynomial, where A(j)(j = 0, 1,...,m) are n x n complex matrice s and lambda is a complex variable. For a q is an element of [0, 1] the q-n umerical range of P(lambda) is defined as W-q[P(lambda)] = (lambda is an element of C : x*P(lambda )y = 0, x*x = y*y = 1 and x*y = q). In this paper we study W-q[P(lambda)] and our emphasis is on the geometrica l properties of W-q[P(lambda)]. We consider the location of W-q[P(lambda)] in the complex plane and a theorem concerning the boundary of W-q[P(lambda] is also obtained.