Elementary inclusion relations for generalized numerical ranges

Abstract

Let γ 1,…,γ n be complex constants. The set W (γ 1,…,γ n ) ( A) = {Σγ j( Ax j, x j)},where ( x 1,…, x n ) vary over all orthonormal systems in C n , is called a generalized numerical range of a given n × n matrix A. In this paper we study inclusion relations of the form W (γ 1,…,γ n ) ⊂λ W (γ′ 1,…,γ′ n ) which hold uniformly for all n-square matrices A In particular we concentrate on the case where the coefficients are real. Such inclusion relations yield simple inequalities among generalized numerical radii. Finally, a further generalization of the above numerical range is discussed.