Computing the numerical range of Krein space operators

Natalia Bebiano; J. da Providência; A. Nata; J.P. da Providência

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page 146-156, electronic only
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Natalia Bebiano, et al. "Computing the numerical range of Krein space operators." Open Mathematics 13.1 (2015): 146-156, electronic only. <http://eudml.org/doc/268706>.

@article{NataliaBebiano2015,
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.},
author = {Natalia Bebiano, J. da Providência, A. Nata, J.P. da Providência},
journal = {Open Mathematics},
keywords = {Indefinite inner product; Krein space; Numerical range; Compression; indefinite inner product; numerical range; compression; Hilbert space; indefinite self-adjoint involution; algorithm; numerical results},
language = {eng},
number = {1},
pages = {146-156, electronic only},
title = {Computing the numerical range of Krein space operators},
url = {http://eudml.org/doc/268706},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Natalia Bebiano
AU - J. da Providência
AU - A. Nata
AU - J.P. da Providência
TI - Computing the numerical range of Krein space operators
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - 146
EP - 156, electronic only
AB - 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.
LA - eng
KW - Indefinite inner product; Krein space; Numerical range; Compression; indefinite inner product; numerical range; compression; Hilbert space; indefinite self-adjoint involution; algorithm; numerical results
UR - http://eudml.org/doc/268706
ER -

References

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