TY - JOUR

T1 - Operators with real parts at least

AU - Gau, Hwa Long

AU - Wu, Pei Yuan

N1 - Publisher Copyright:
© 2016 Informa UK Limited, trading as Taylor & Francis Group.

PY - 2017/10/3

Y1 - 2017/10/3

N2 - 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.

AB - 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.

KW - S-matrix

KW - numerical range

UR - http://www.scopus.com/inward/record.url?scp=85004147062&partnerID=8YFLogxK

U2 - 10.1080/03081087.2016.1267106

DO - 10.1080/03081087.2016.1267106

M3 - 期刊論文

AN - SCOPUS:85004147062

SN - 0308-1087

VL - 65

SP - 1988

EP - 1999

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

IS - 10

ER -