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

Let A = ( A 11 A 12 A 21 A 22 ) ∈ M n , where A 11 ∈ M m with m ⩽ n / 2 , be such that the numerical range of A lies in the set { e i φ z ∈ C : | ℑ z | ⩽ ( ℜ z ) tan α } , for some φ ∈ [ 0 , 2 π ) and α ∈ [ 0 , π / 2 ) . We obtain the optimal containment region for the generalized eigenvalue λ satisfying λ ( A 11 0 0 A 22 ) x = ( 0 A 12 A 21 0 ) x for some nonzero x ∈ C 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 ) | ⩽ sec 2 m ( α ) × | det ( A 11 ) det ( A 22 ) | . In particular, if A is an accretive-dissipative matrix, then | det ( A ) | ⩽ 2 m | det ( A 11 ) det ( A 22 ) | . These affirm some conjectures of Drury and Lin.