Operators with real parts at least

Hwa Long Gau, Pei Yuan Wu

Research output: Contribution to journalArticlepeer-review


For an Sn-matrix (n ≥ 3) A (a contraction with eigenvalues in the open unit disc and rank (ln = A*A) = 1), we consider the numerical range properties of B = A(ln - 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,...,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)ln 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.

Original languageEnglish
Pages (from-to)1988-1999
Number of pages12
JournalLinear and Multilinear Algebra
Issue number10
StatePublished - 3 Oct 2017


  • S-matrix
  • numerical range


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