Type: Article
Publication Date: 1977-03-01
Citations: 24
DOI: https://doi.org/10.2140/pjm.1977.69.45
Let c=(γ lf •• ,τ' n ) be given.The generalized numerical range of an nXn matrix A, associated with c, is the set W c (A)={Σγj(Axj, Xj)} where (x l9 •••,#") varies over orthonormal systems in C n .Characterizations of this range, for real c, are given.Next, we study integrals of the form \W c (A)dμ(c) where μ(c) is a measure defined on a domain in R n .The above characterizations are used to study the inclusion [w c (A)dμ(c)c:λW c >(A).We determine those λ, for which this inclusion holds for all nXn 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 W c (A)a λWAA) which hold for all A. 1* Introduction* The generalized numerical range of an n x n complex matrix A, associated with a fixed vector c = (Ύ 19 , 7 W ) e C\ is the set of complex numbers