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## Abstract

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 ∈ ℂ^{n}: (Bx, x) = λǁxǁ^{2}}, is a subspace of dimension one with the property that x, Bx,...,B^{n-1}x 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.

Original language | English |
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Pages (from-to) | 1988-1999 |

Number of pages | 12 |

Journal | Linear and Multilinear Algebra |

Volume | 65 |

Issue number | 10 |

DOIs | |

State | Published - 3 Oct 2017 |

## Keywords

- S-matrix
- numerical range

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