Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access November 20, 2014

Computing the numerical range of Krein space operators

  • Natalia Bebiano EMAIL logo , J. da Providência , A. Nata and J.P. da Providência
From the journal Open Mathematics


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.


[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. Search in 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. 10.1080/0308108031000134981Search in 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. 10.1016/j.laa.2005.05.021Search in 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. Search in 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. 10.1016/j.laa.2008.01.017Search in Google Scholar

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

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

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

[12] R.A. Horn and C.R. Johnson, Matrix Analysis. Cambridge University Press, New York, 1985. 10.1017/CBO9780511810817Search in Google Scholar

[13] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis. Cambridge University Press, Cambridge, 1991. 10.1017/CBO9780511840371Search in 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. 10.13001/1081-3810.1013Search in 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. 10.1090/S0002-9939-98-04242-7Search in Google 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. Search in Google Scholar

[17] M. Marcus and C. Pesce, Computer generated numerical ranges and some resulting theorems. Linear and Multilinear Algebra, 20 (1987), 121–157. 10.1080/03081088708817748Search in Google Scholar

[18] P.J. Psarrakos, Numerical range of linear pencils, Linear Algebra Appl. 317 (2000), 127-141. 10.1016/S0024-3795(00)00145-2Search in 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), 554-567.10.1080/03081087.2013.779269Search in Google Scholar

Received: 2014-1-2
Accepted: 2014-11-5
Published Online: 2014-11-20

© 2015 Natalia Bebiano et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 28.2.2024 from
Scroll to top button