A Study on Operators with Real Parts at Least - I

For an &-matrix (n^3) A (a contraction with eigenvalues in the open unit disc and rank(/n-A*A)=1), we consider the numerical range properties of B=A(/n-A)-1. We want to show that W(B), the numerical range of B，is contained in the half-plane Re(z)^-l/2, its boundary ^W(B) contains exactly one line segment Z, which lies on Re(z)=-l/2, and, for any X in 5W(B)\L, M = {x 6 C^Bx’x) = A||at||2} is a sub space of dimension one with the property that x, Bx,…Bn lx are linearly independent for any nonzero vector x inM. Using such properties, we conjecture that any n-by-n matrix C with Re(C)^(-1/2)/n can be extended, under unitary similarity, to a direct sum D㊉B㊉…㊉B of a diagonal matrix D with diagonals on the line Re(z)=-l/2 and copies of B of the above type, and, moreover, if 5W(C) has a common point with ^W(B)\L, then C has B as a direct summand. This will generalize previous results of the author’s for a nilpotent C.