Operators with real parts at least
Hwa-Long Gau, Pei Yuan Wu · Linear and Multilinear Algebra · 2016
For an -matrix () A (a contraction with eigenvalues in the open unit disc and ), we consider the numerical range properties of . It is shown that W(B), the numerical range of B, is contained in the half-plane , its boundary contains exactly one line segment L, which lies on , and, for any in , is a subspace of dimension one with the property that are linearly independent for any nonzero vector x in M. Using such properties, we prove that any n-by-n matrix C with can be extended, under unitary similarity, to a direct sum of a diagonal matrix D with diagonals on the line and copies of B of the above type, and, moreover, if has a common point with , then C has B as a direct summand. This generalizes previous results of the authors for a nilpotent C.