Joint zero sets and ranges of several Hermitian forms over complex and quaternionic scalars (Q1827498)

The aim of the paper is to study some problems concerning the connectedness and convexity of some sets associated with a vectorial Hermitian form \(h:V\to \mathbb R^{n}\) (each component is a Hermitian form) on an inner product space \(V\) over \(\mathbb F=\mathbb C\) or \(\mathbb H\), the quaternions. The problems that are studied are related to classical works of \textit{O. Toeplitz} [Math. Z. 2, 187--197 (1918; JFM 46.0157.02)] and \textit{F. Hausdorff} [Math. Z. 3, 314--316 (1919; JFM 47.0088.02)]. One denotes as \(V_{+}\) and \(V_{1}\) the subsets of \(V\) of nonzero and norm one vectors respectively and by \(Z=h^{-1}(0)\), \(Z_{+}=V_{+}\cap Z\) and \(Z_{1}=V_{1}\cap Z\). The path connectedness of the sets \(Z\), \(Z_{+}\) and \(Z_{1}\) is studied. Denoting by \(d\) the dimension of \(\mathbb F\) as a real space, the authors prove that for \(\mathbb F=\mathbb C\) or \(\mathbb H\), if \(n<d\) or \(n\leq d\), provided that \(\dim V>2\), then \(Z_{+}\) and \(Z_{1}\) are path connected. The case \(\mathbb F=\mathbb C\) is known from [\textit{Y. Lyubich} and \textit{A. Markus}, Positivity 1, No. 3, 239--254 (1997; Zbl 0891.15023)], but the proof in the paper under review is simpler. The authors give also more elementary proofs contained in [\textit{P. Binding}, Proc. Am. Math. Soc. 94, 581--584 (1985; Zbl 0572.55014)] of the fact that the joint range \(W_{1}=h(V_{1})\) is convex for \(n\leq d\) and for \(n\leq d+1\), provided that \(\dim V>2\). The case \(\dim V=1\) is also studied by the authors and some examples concerning the optimality of their results are given.