Determinantal and Eigenvalue Inequalities for Matrices with Numerical Ranges in a Sector

Let A = ( A11 A12 A21 A22 ) ∈ Mn, where A11 ∈ Mm with m ≤ n/2, be such that the numerical range of A lies in the set {eiφz ∈ C : |=z| ≤ (<z) tanα}, for some φ ∈ [0, 2π) and α ∈ [0, π/2). We obtain the optimal containment region for the generalized eigenvalue λ satisfying λ ( A11 0 0 A22 ) x = ( 0 A12 A21 0 ) x for some nonzero x ∈ C, and the optimal eigenvalue containment region of the matrix Im−A 11 A12A −1 22 A21 in case A11 and A22 are invertible. From this result, one can show | det(A)| ≤ sec(α)|det(A11) det(A22)|. In particular, if A is a accretive-dissipative matrix, then | det(A)| ≤ 2m| det(A11) det(A22)|. These affirm some conjectures of Drury and Lin. AMS subject classifications. 15A45.