A generalized numerical range: the range of a constrained sesquilinear form

Abstract

Let Mn denote the algebra of all nxn complex matrices. For a given q∊C with ∣Q∣≤1, we define and denote the q-numerical range of A∊Mn by Wq (A)={x ∗ Ay:x,y∊C n , x ∗ x−y ∗ y=1,x ∗ y=q } The q-numerical radius is then given by rq (A)=sup{∣z∣:z∊W q (A)}. When q=1,W q (A) and r q (A) reduce to the classical numerical range of A and the classical numerical radius of A, respectively. when q≠0, another interesting quantity associated with W q (A) is the inner q-numerical radius defined by [rtilde] q (A)=inf{∣z∣:z∊W q (A)} In this paper, we describe some basic properties of W q (A), extending known results on the classical numerical range. We also study the properties of rq considered as a norm (seminorm if q=0) on Mn .Finally, we characterize those linear operators L on Mn that leave Wq ,rq of [rtilde]q invariant. Extension of some of our results to the infinite dimensional case is discussed, and open problems are mentioned.