Determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector

Abstract

Let $A = \pmatrix A_{11} & A_{12} \cr A_{21} & A_{22}\cr\pmatrix \in M_n$, where $A_{11} \in M_m$ with $m \le n/2$, be such that the numerical range of $A$ lies in the set $\{e^{iφ} z \in \IC: |\Im z| \le (\Re z) \tan α\}$, for some $φ\in [0, 2π)$ and $α\in [0, π/2)$. We obtain the optimal containment region for the generalized eigenvalue $λ$ satisfying $$λ\pmatrix A_{11} & 0 \cr 0 & A_{22}\cr\pmatrix x = \pmatrix 0 & A_{12} \cr A_{21} & 0\cr\pmatrix x \quad \hbox{for some nonzero} x \in \IC^n,$$ and the optimal eigenvalue containment region of the matrix $I_m - A_{11}^{-1}A_{12} A_{22}^{-1}A_{21}$ in case $A_{11}$ and $A_{22}$ are invertible. From this result, one can show $|\det(A)| \le \sec^{2m}(α) |\det(A_{11})\det(A_{22})|$. In particular, if $A$ is a accretive-dissipative matrix, then $|\det(A)| \le 2^m |\det(A_{11})\det(A_{22})|$. These affirm some conjectures of Drury and Lin.