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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 halfplane 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^{n1}x are linearly independent for any nonzero vector x in M. Using such properties, we prove that any nbyn 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 

Pages (fromto)  19881999 
Number of pages  12 
Journal  Linear and Multilinear Algebra 
Volume  65 
Issue number  10 
DOIs  
State  Published  3 Oct 2017 
Keywords
 Smatrix
 numerical range
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Dive into the research topics of 'Operators with real parts at least'. Together they form a unique fingerprint.Projects
 1 Finished

A Study on ZeroDilation Index of SnMatrix and Companion Matrix
Gau, H.L. (PI)
1/08/16 → 31/07/17
Project: Research