The product field of values
The authors consider the following special generalization of the classical numerical range \[ P(A,B)=\{ \langle Ax,x\rangle \langle Bx, x\rangle \;:\;x\in {\mathbb C}^n,~ \|x\|=1\}, \] which they call the product field of values. Basic properties of the product field of values are established. The main results are Theorem 1: For \(A,B\in M_n(\mathbb{C})\), \(P(A,B)\) is ray convex and therefore simply convex. (Note that a connected set \(\Omega\) is ray convex if for each \(\alpha\notin\Omega\) there is a ray emanating from \(\alpha\) not intersecting \(\Omega\).) and Theorem 2: The interior of \(P(A,B)\) is empty if and only if one of the following conditions hold: (i) both \(A\) and \(B\) are scalar multiples of Hermitian matrices, (ii) both \(A\) and \(B\) are linear combinations of the identity matrix and some Hermitian matrix \(H\), and (iii) \(A\) and \(B^*\) are linearly dependent.
scientific article; zbMATH DE number 6138404