Computing the numerical range of Krein space operators

• Autorzy: Natalia Bebiano , J. da Providência , A. Nata , J.P. da Providência
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Indefinite inner product; Krein space; Numerical range; Compression

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2015
• Tom: 13
• Numer: 1

Daty

otrzymano

2014-01-02

zaakceptowano

2014-11-05

online

2014-11-20

Twórcy

autor

Natalia Bebiano

• CMUC, University of Coimbra, Department of Mathematics, P 3001-454 Coimbra,
  Portugal

autor

J. da Providência

• University of Coimbra, Department of Physics, P 3004-516 Coimbra, Portugal

autor

A. Nata

• CMUC, Polytechnic Institute of Tomar, Department of Mathematics, P 2300-313 Tomar, Portugal

autor

J.P. da Providência

• Depatamento de Física, Univ. of Beira Interior, P-6201-001 Covilhã, Portugal

Bibliografia

• [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.
• [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.
• [3] N. Bebiano, H. Nakazato, J. da Providência, R. Lemos and G. Soares, Inequalities for JHermitian matrices, Linear Algebra Appl. 407 (2005) 125–139.
• [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.
• [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. [WoS]
• [6] N. Bebiano, I. Spitkovsky, Numerical ranges of Toeplitz operators with matrix symbols, Linear Algebra Appl., 436 (2012) 1721–1726. [WoS]
• [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.
• [8] M.-T. Chien and H. Nakazato, The numerical range of a tridiagonal operator, J. Math. Anal. Appl., 373, No. 1 (2011), 297–304.
• [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. [WoS]
• [10] C. Crorianopoulos, P. Psarrakos and F. Uhlig. A method for the inverse numerical range problem. Linear Algebra Appl. 24 (2010) 055019.
• [11] I.Gohberg, P.Lancaster and L.Rodman, Matrices and Indefinite Scalar Product. Birkhäuser, Basel-Boston, 1983.
• [12] R.A. Horn and C.R. Johnson, Matrix Analysis. Cambridge University Press, New York, 1985.
• [13] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis. Cambridge University Press, Cambridge, 1991.
• [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.
• [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. [Crossref]
• [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.
• [17] M. Marcus and C. Pesce, Computer generated numerical ranges and some resulting theorems. Linear and Multilinear Algebra, 20 (1987), 121–157.
• [18] P.J. Psarrakos, Numerical range of linear pencils, Linear Algebra Appl. 317 (2000), 127-141.
• [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.

Identyfikatory

DOI

10.1515/math-2015-0014