Generalization of numerical range of polynomial operator matrices

Suppose that  is a polynomial matrix operator where  for , are  complex matrix and let  be a complex variable. For an  Hermitian matrix , we define the -numerical range of polynomial matrix of  as , where . In this paper we study   and our emphasis is on the geometrical properties of . We consider the location of   in the complex plane and  a theorem concerning  the boundary of is also obtained.  Possible generalazations of our results including their extensions to bounded linerar operators on an infinite dimensional Hilbert space are described.