Symmetry of the spatial numerical ranges of algebra polynomials and of matrix polynomials (Q1966330)

Let \(E\) be a unital \(*\)-algebra and \(\varphi: E\to{\mathcal L}(H_\varphi)\) a \(*\)-representation. For a polynomial \(P\) with coefficients in \(E\) the \(\varphi\)-spatial numerical range is defined as the set of those complex numbers \(\lambda_0\) such that \((\varphi(P(\lambda_0)) \xi|\xi)= 0\) for some \(\xi\in H_\varphi\) with \((\xi|\xi)= 1\). In this setting and for unital \(lmc\) \(*\)-algebras, where the Bonsall numerical range is considered, conditions are given to imply the numerical range or the union of two numerical ranges being symmetric to the real axis or to a line through the origin.