Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils
For an \(n\)-by-\(n\) complex matrix \(A\) and \(1\leq k\leq n\) the \(k\)th numerical range of \(A\) is defined by \(W_k(A)=\{(1/k) \sum^k_{j=1}x^*_jAx_j: \{x_1,\dots,x_n\}\) is an orthonormal set in \({\mathbb C}^n\}\). When \(k=1\), \(W_k(A)\) reduces to the classical numerical range. It is known that \(\{\text{tr}A/n\}= W_n(A)\subseteq W_{n-1}(A)\subseteq\dots\subseteq W_1(A)\). The purpose of this paper is to study conditions on \(A\) so that \(W_m(A)=W_k(A)\) for some given \(m,k\). It turns out that this study is related to a conjecture of \textit{R. Kippenhahn} [Math. Nachr. 6, 193-228 (1951; Zbl 0044.16201)]. As a by-product the authors obtain a new class of \(6\times 6\) counterexamples to the Kippenhahn conjecture.
scientific article; zbMATH DE number 1109507