Let c = (γ1,⋯,γn) be given.
The generalized numerical range of an n × n matrix A, associated with c, is the set
Wc(A) = {∑γj(Axj,xj)} where (x1,⋯,xn) varies over orthonormal systems in
Cn. Characterizations of this range, for real c, are given. Next, we study
integrals of the form ∫Wc(A)dμ(c) where μ(c) is a measure defined on a
domain in Rn. The above characterizations are used to study the inclusion
∫Wc(A)dμ(c) ⊂ λWc′(A). We determine those λ, for which this inclusion holds for
all n×n matrices A. Such relations lead to more elementary ones, when the integral
reduces to a finite linear combination of ranges. In particular, we obtain
the inclusion relations of the form Wc(A) ⊂ λWc′(A) which hold for all
A.
