Flat portions on the boundary of the indefinite numerical range of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mrow><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math> matrices

Bebiano, Natália; Providência, J. da; Teixeira, Ricardo Emanuel Cunha

doi:10.1016/j.laa.2008.01.017

Cited by 2 publications

(2 citation statements)

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“…The investigation of the numerical range and its generalizations has attracted the interest of researchers from different areas of pure and applied science [4,10], since the concept was born with Toeplitz [26] and Hausdorff [19] in the second decade of the last century. The numerical range of linear operators in an indefinite inner product space is a topic which has deserved the study by some authors; we refer the interested reader to [3,5,6,11,13,14,22,23,24,25]. Ilya Spitkovsky and collaborators published several inspiring papers on the ellipticity of the classical numerical range (see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…All the possible shapes of the numerical range W J (A) of matrices of size 3 were classified in [13,14] based on the factorability of the KNR generating polynomial p A . If this polynomial factors into three linear factors, then C J (A) consists of three points, which are the eigenvalues of A, and W J (A) is the pseudo-convex hull of the eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

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Computing the numerical range of Krein space operators

et al. 2014

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Consider the Hilbert space (H; h ; i) equipped with the indefinite inner product OEu; v D v J u, u; v 2 H, where J is an indefinite self-adjoint involution acting on H. The Krein space numerical range W J .T / of an operator T acting on H is the set of all the values attained by the quadratic form OET u; u, with u 2 H satisfying OEu; u D˙1. We develop, implement and test an alternative algorithm to compute W J .T / in the finite dimensional case, constructing 2 by 2 matrix compressions of T and their easily determined elliptical and hyperbolical numerical ranges. The numerical results reported here indicate that this method is very efficient, since it is faster and more accurate than either of the existing algorithms. Further, it may yield easy solutions for the inverse indefinite numerical range problem. Our algorithm uses an idea of Marcus and Pesce from 1987 for generating Hilbert space numerical ranges of matrices of size n.

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