The G-bilinear range

Abstract

Let 1 ⩽ r⩽ n A a linear transformation on an n-dimensional unitary space V and let G = [G₁:G₂:G₃]bean r × 3 r matrix.A collection of 2r vectors x₁,…,x_r,y₁,…,y_rin V me said to be G vectors whenever G₁= [(x_i, X_j)], G₂ = [y_i,y_j], and G₃= [(X_i,y_j)]. In the present paper we examine the structure of the setW(A;G), of values of the form (Ax₁,y₁)+ ⋯ +(Ax_r,y_r) where x₁,…,x_r,y₁,…,y_r run over all G-vectors. Specifically, conditions for W(A;G) to be convex, the origin, or empty are given along with some upper bounds on the maximum modulus of any element in W(A;G). These results extend some of the classical results of Hausdorff, Toeplitz, von Neumann, Fan, Berger, and Westwick concerning higher numerical ranges.

† The research of both authors was supported by the Air Force Office of Scientific Research under contract AFOSR 79-0127.

† The research of both authors was supported by the Air Force Office of Scientific Research under contract AFOSR 79-0127.

Notes

† The research of both authors was supported by the Air Force Office of Scientific Research under contract AFOSR 79-0127.