Operators with real parts at least

For an S_n-matrix (n ≥ 3) A (a contraction with eigenvalues in the open unit disc and rank (l_n = A*A) = 1), we consider the numerical range properties of B = A(l_n - A)^-1. It is shown that W(B), the numerical range of B, is contained in the half-plane Rez ≥ -1/2, its boundary ∂W(B) contains exactly one line segment L, which lies on Re z = -1/2, and, for any λ in ∂W(B) \ L, M ≡ {x ∈ ℂⁿ: (Bx, x) = λǁxǁ²}, is a subspace of dimension one with the property that x, Bx,...,B^n-1x are linearly independent for any nonzero vector x in M. Using such properties, we prove that any n-by-n matrix C with Re C ≥ (-1/2)l_n can be extended, under unitary similarity, to a direct sum D⊕B⊕...⊕B of a diagonal matrix D with diagonals on the line Rez = -1/2 and copies of B of the above type, and, moreover, if ∂W(C) has a common point with ∂W(B)\L, then C has B as a direct summand. This generalizes previous results of the authors for a nilpotent C.