Higher rank numerical hulls of matrices (Q2885136)

For any \(n\)-by-\(n\) complex matrix \(A\) and any positive integer \(k\), the rank-\(k\) numerical range \(\Lambda_k(A)\) and rank-\(k\) spectrum \(\sigma_k(A)\) of \(A\) are defined by NEWLINE\[NEWLINE\Lambda_k(A)= \{\lambda\in\mathbb{C}: X*AX=\lambda_k\text{ for some \(n\)-by-\(k\) matrix \(X\) with \(X*X= I_k\)}\}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\sigma_k(A)= \{\lambda\in\mathbb{C}: \dim\ker(\lambda I_n-A)\geq k\},NEWLINE\]NEWLINE respectively. In recent years, the former has been studied intensely by numerical analysis. When \(k= 1\), \(\Lambda_1(A)\) and \(\sigma_1(A)\) are just the classical numerical range \(W(A)\) and spectrum \(\sigma(A)\) of \(A\).NEWLINENEWLINE In the present paper, the author introduces, for any positive integer \(m\), the more general rank-\(k\) numerical hull \(X^m_k(A)\) of \(A\), namely, NEWLINE\[NEWLINEX^m_k(A)\equiv \{\lambda\in\mathbb{C}: (\lambda,\lambda^2,\dots, \lambda^m)\in \text{conv}(\Lambda_k(A,A^2,\dots, A^m))\},NEWLINE\]NEWLINE where \(\text{conv}(\Lambda_k(A, A^2,\dots, A^m))\) denotes the convex hull of the joint rank-\(k\) numerical range \(\Lambda_k(A, A^2,\dots, A^m)\) of J\((A,A^2,\dots, A^m)\): NEWLINE\[NEWLINE\Lambda_k(A, A^2,\dots, A^m)\equiv\{(\lambda_1,\dots, \lambda_m)\in \mathbb{C}^m: X* A^j X= \lambda_j I_k\text{ for some \(n\)-by-\(k\) matrix \(X\)}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\text{with \(X* X= I_k\) for all \(j,1\leq j\leq m\)}\}.NEWLINE\]NEWLINE Among other things, it is shown that if \(A\) is Hermitian and \(m\geq 2\), then \(X^m_k(A)= \sigma_k(A)\), and if \(A\) is unitary of size \(2k\) and \(k\), \(m\geq 2\), then \(X^m_k(A)=\emptyset\).