Operators with real parts at least-1/2

Abstract

For an Sn-matrix (n >= 3) A (a contraction with eigenvalues in the open unit disc and rank (In -A* A) = 1), we consider the numerical range properties of B = A(In -A)-1. It is shown that W(B), the numerical range of B, is contained in the half-plane Re z = -1/2, its boundary. W(B) contains exactly one line segment L, which lies on Re z = -1/2, and, for any. in. W(B) \ L, M = {x. Cn : similar to Bx, x similar to =. similar to x similar to 2} is a subspace of dimension one with the property that x, Bx,..., Bn-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) In can be extended, under unitary similarity, to a direct sumD. B. similar to similar to similar to. B of a diagonal matrix D with diagonals on the line Re z = -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.