Numerical range circumscribed by two polygons
The main result of this paper asserts: If \(a_1,a'_1\), \(a_2,a'_2\), \dots, \(a_{n+1},a'_{n+1}\) are \(2n+2\) distinct points on the unit circle, then there is an \(\mathcal S_n\)-matrix \(A\in \mathbb{C}^{n\times n}\), unique up to unitary equivalence, with numerical range \(W(A)\) circumscribed by the two polygons \(a_1,a_2,\dots,a_{n+1}\) and \(a'_1,a'_2,\dots,a'_{n+1}\). Here \(A\in\mathcal S_n\) if \(A\) is a contraction (its operator norm is at most 1), all eigenvalues of \(A\) are in the open unite disc, and \(\text{rank}(I_n-A^*A)=1\). The results in this paper were inspired by an interesting projective geometry theorem assuring the existence of a conical curve circumscribed by two triangles which are inscribed on another conical curve. This research completes and generalizes the results obtained recently by the authors [Linear and Multilinear Algebra 45, 49--73 (1998; Zbl 0918.15008), ibid. 45, 359--373 (1999; Zbl 0942.15019), ibid. 52, 195--201 (2004; Zbl 1066.15027) and Taiwanese J. Math. 7, 173--193 (2003; Zbl 1051.15019)].
scientific article; zbMATH DE number 2081906