Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year


IMPACT FACTOR 2017: 0.831
5-year IMPACT FACTOR: 0.836

CiteScore 2017: 0.68

SCImago Journal Rank (SJR) 2017: 0.450
Source Normalized Impact per Paper (SNIP) 2017: 0.829

Mathematical Citation Quotient (MCQ) 2017: 0.32

ICV 2017: 161.82

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 13, Issue 1

Issues

Volume 13 (2015)

Computing the numerical range of Krein space operators

Natalia Bebiano / J. da Providência / A. Nata / J.P. da Providência
Published Online: 2014-11-20 | DOI: https://doi.org/10.1515/math-2015-0014

Abstract

Consider the Hilbert space (H,〈• , •〉) equipped with the indefinite inner product[u,v]=v*J u,u,v∈ H, where J is an indefinite self-adjoint involution acting on H. The Krein space numerical range WJ(T) of an operator T acting on H is the set of all the values attained by the quadratic form [Tu,u], with u ∈H satisfying [u,u]=± 1. We develop, implement and test an alternative algorithm to compute WJ(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.

Keywords : Indefinite inner product; Krein space; Numerical range; Compression

References

  • [1] Y.H. Au-Yeung and N.K. Tsing, An extension of the Hausdorff-Toeplitz theorem on the numerical range, Proc. Amer. Math. Soc., 89 (1983) 215–218. Google Scholar

  • [2] N. Bebiano, R. Lemos, J. da Providência and G. Soares, On generalized numerical ranges of operators on an indefinite inner product space, Linear and Multilinear Algebra 52 No. 3–4, (2004) 203–233. Google Scholar

  • [3] N. Bebiano, H. Nakazato, J. da Providência, R. Lemos and G. Soares, Inequalities for JHermitian matrices, Linear Algebra Appl. 407 (2005) 125–139. Google Scholar

  • [4] N. Bebiano, J. da Providência, A. Nata and G. Soares, Krein Spaces Numerical Ranges and their Computer Generation, Electron. J. Linear Algebra, 17 (2008) 192–208. Google Scholar

  • [5] N. Bebiano, J. da Providência, R. Teixeira, Flat portions on the boundary of the indefinite numerical range of 3 x 3 matrices, Linear Algebra Appl. 428 (2008) 2863-2879. Web of ScienceGoogle Scholar

  • [6] N. Bebiano, I. Spitkovsky, Numerical ranges of Toeplitz operators with matrix symbols, Linear Algebra Appl., 436 (2012) 1721–1726. Web of ScienceGoogle Scholar

  • [7] N. Bebiano, J. da Providência, A. Nata and J. P. da Providência, An inverse problem for the indefinite numerical range, Linear Algebra Appl. to appear. Google Scholar

  • [8] M.-T. Chien and H. Nakazato, The numerical range of a tridiagonal operator, J. Math. Anal. Appl., 373, No. 1 (2011), 297–304. Google Scholar

  • [9] C.F. Dunkl, P. Gawron, J.A. Holbrook, Z. Puchala and K. Zyczkowski, Numerical shadows: measures and densities of numerical range, Linear Algebra Appl. 434 (2011) 2042–2080. Web of ScienceGoogle Scholar

  • [10] C. Crorianopoulos, P. Psarrakos and F. Uhlig. A method for the inverse numerical range problem. Linear Algebra Appl. 24 (2010) 055019. Google Scholar

  • [11] I.Gohberg, P.Lancaster and L.Rodman, Matrices and Indefinite Scalar Product. Birkhäuser, Basel-Boston, 1983. Google Scholar

  • [12] R.A. Horn and C.R. Johnson, Matrix Analysis. Cambridge University Press, New York, 1985. Google Scholar

  • [13] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis. Cambridge University Press, Cambridge, 1991. Google Scholar

  • [14] C.-K. Li and L. Rodman, Shapes and computer generation of numerical ranges of Krein space operators. Electron. J. Linear Algebra, 3 (1998) 31–47. Google Scholar

  • [15] C.-K. Li and L. Rodman, Remarks on numerical ranges of operators in spaces with an indefinite metric, Proc. Amer. Math. Soc. 126 No. 4, (1998) 973–982. CrossrefGoogle Scholar

  • [16] C.-K. Li, N.K. Tsing and F. Uhlig. Numerical ranges of an operator on an indefinite inner product space. Electron. J. Linear Algebra 1 (1996) 1–17. Google Scholar

  • [17] M. Marcus and C. Pesce, Computer generated numerical ranges and some resulting theorems. Linear and Multilinear Algebra, 20 (1987), 121–157. Google Scholar

  • [18] P.J. Psarrakos, Numerical range of linear pencils, Linear Algebra Appl. 317 (2000), 127-141. Google Scholar

  • [19] F. Uhlig, Faster and more accurate computation of the field of values boundary for n by n matrices, Linear and Multilinear Algebra 62(5) (2014), 554-567.Google Scholar

About the article

Received: 2014-01-02

Accepted: 2014-11-05

Published Online: 2014-11-20


Citation Information: Open Mathematics, Volume 13, Issue 1, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0014.

Export Citation

© 2015 Natalia Bebiano et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in