Elementary inclusion relations for generalized numerical ranges (Q1239209)

Let \(\gamma_1,\dots,\gamma_n\) be complex constants. The set \(W_{(\gamma_1,\dots,\gamma_n)}(A) =\{\sum\gamma_j(Ax_j,x_j)\}\) where \((x_1,\dots,x_n)\) vary over all orthonormal systems in \(\mathbb C ^n\), is called a generalized numerical range of a given \(n\times n\) matrix \(A\). In this paper we study inclusion relations of the form \(W_{(\gamma_1,\dots,\gamma_n)}\subset\lambda W_{(\gamma_1',\dots,\gamma_n')}\) which hold uniformly for all \(n\)-square matrices \(A\). In particular we concentrate on the case where the coefficients are real. Such inclusion relations yields simple inequalities among generalized numerical radii. Finally, a further generalization of the above numerical range is discussed.