Birkhoff-James approximate orthogonality sets and numerical ranges
For any \(n\)-by-\(m\) matrices \(A\) and \(B\) with \(B\neq 0\), any matrix norm \(\|.\|\) and any \(\varepsilon\), \(0\leq\varepsilon< 1\), the Birkhoff-James \(\varepsilon\)-orthogonality set of \(A\) with respect to \(B\) is defined as \[ F^\varepsilon_{\|.\|}(A; B)= \{u\in\mathbb{C}:\| A-\lambda B\|\geq (1-\varepsilon^2)^{1/2}\| B\|\,|u-\lambda|\text{ for all }\lambda\text{ in }\mathbb{C}\}. \] This is a generalization of the classical numerical range (when \(n= m\), \(\|.\|\) is the spectral norm, \(\varepsilon= 0\) and \(B= I_n\)) and the \(q\)-numerical range (when \(n= m\), \(\|.\|\) and \(B\) as above, and \(\varepsilon= q\)) of square matrices. This approach in various forms has also been adopted by other people in recent years. The above-defined sets are easily seen to be nonempty, compact and convex. Other basic properties compatible with the classical case are proven in Section 2 of this paper. Then in Section 3, the Birkhoff-James \(\varepsilon\)-orthogonality set of a matrix polynomial \[ p(z)= A_\ell z^\ell+ A_{\ell-1} z^{\ell-1}+\cdots+ A_1 z+ A_0, \] where the \(A_j\)'s are \(n\)-by-\(m\) matrices with \(A_\ell\neq 0\), with respect to \(B\neq 0\) is also defined by \[ W^\varepsilon_{\|.\|}(p(z); B)= \{u\in\mathbb{C}:\| p(u)-\lambda B\|\geq (1-\varepsilon^2)^{1/2}\| B\|\, |\lambda|\text{ for all }\lambda\text{ in }\mathbb{C}\}, \] for any matrix norm \(\|.\|\) and any \(\varepsilon\), \(0\leq\varepsilon< 1\), which generalizes the standard numerical range and \(q\)-numerical range of \(p(z)\). Section 4 then treats properties of the boundary points of these \(\varepsilon\)-orthogonality sets. In particular, it is shown that if \(u_0\) is in \(\partial W^\varepsilon_{\|.\|}(p(z); B)\), then \(0\) is in \(\partial F^\varepsilon_{\|.\|}(p(u_0); B)\). The paper concludes in Section 5 with some simplified expressions for both \(\varepsilon\)-orthogonality sets when the matrix norm \(\|.\|\) is induced by an inner product of matrices.
scientific article; zbMATH DE number 5867199
scientific article