Vol. 69, No. 1, 1977

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On characterizations and integrals of generalized numerical ranges

Moshe Goldberg and Ernst Gabor Straus

Vol. 69 (1977), No. 1, 45–54

Let c = (γ1,n) 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)(c) where μ(c) is a measure defined on a domain in Rn. The above characterizations are used to study the inclusion Wc(A)(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.

Mathematical Subject Classification 2000
Primary: 47A10
Secondary: 15A60
Received: 2 August 1976
Revised: 29 October 1976
Published: 1 March 1977
Moshe Goldberg
Ernst Gabor Straus