Compressions and direct sums

For an n-by-n matrix A partitioned as [BCDE] with B of size m (1 ≤ m< n ), we consider various conditions on A and B which guarantee that C= 0 and D= 0 . Such conditions involve eigenvalues, Kippenhahn polynomials or numerical ranges of A and B. One of them says that if ‖ A‖ = 1 , B is of class S_m , and W(A) = W(B) , then we always have C= 0 and D= 0 . There are others involving A, B, and E: if, for some θ₁ and θ₂ in [ 0 , 2 π) with | θ₁- θ₂| ≠ 0 , π , the spectrum of Re(eiθjA) equals the union of the spectra of Re(eiθjB) and Re(eiθjE) for both j= 1 and 2, then C= 0 and D= 0 .