Numerical Range of Left Invariant Lorentzian Metrics on the Heisenberg Group H₃ of Dimension Three

Abstract

The study of eigenvalues and numerical range appears in diffrent scientific fields. We can cite for example the domain of physics, spectral theory, the stability of dynamics electricity, the quantum mechanics. In this paper, we find the spectrum, pseudospectrum and numerical range of left invariant Lorentzian metrics on the Heisenberg group H3 of dimension three. An example is given for metrics g1 and g2 while the second example is provided to support g3.

Keywords

• Heisenberg group
• Lorentz metric
• Bounded operator
• Spectrum
• Pseudo-spectrum
• Numerical range

References

1. [1] C. Van Loan, ”A study of the matrix exponential, Numerical Analysis Report No. 10,” University of Manchester, UK, August, 1975, Reissued as MIMS EPrint, Manchester Institute for Mathematical Sciences, The University of Manchester, UK, November 2006.
2. [2] L. Reichel, L. N .Trefethen, ”Eigenvalues and pseudo-eigenvalues of Toeplitz Matrice,” Lin. Alg. Applics. 162-164, pp. 153-185, 1992.
3. [3] L. Trefethen, M. Embree, ”Spectra and Pseudospectra :The Behavior of Non-Normal Matrices and Operators,” Princeton University Press, Princeton, 2005.
4. [4] T. Ando and C-K. Li (Special editors), ”The Numerical Range and Numerical Radius,” Linear and Multilinear Algebra, vol. 37, no. 1–3, 1994.
5. [5] T. Bayasgalan, ”The numerical range of linear operators in spaces with an in-definite metric (Russian),” Acta Math. Hungar. vol. 57, pp. 157-168, 1991.
6. [6] F. F. Bonsall and Duncan, ”Numerical ranges, Vol. I and II,” Cambridge University Press, 1971 and 1973.
7. [7] F. F. Bonsall and Duncan, ”Studies in Functional Analysis – Numerical ranges,” Studies in Mathematics vol. 21, The Mathematical Association of America, 1980.
8. [8] D. Farenick, ”Matricial Extensions of the numerical range: A brief survey,” Linear and Multilinear Algebra, vol. 34, pp. 197-211, 1993.

9. [9] M.M. Khorami, F. Ershad and B. Yousefi, ”On the Numerical Range of some Bounded Operators,” Journal of Mathematical Extension, vol. 15, 2020.
10. [10] S. Rahmani, ”Metriques de Lorentz sur les groupes de Lie unimodulaires de dimension 3,” J. Geom. Phys., ´ vol. 9, pp. 295-302, 1992.
11. [11] N. Rahmani, S. Rahmani, ”Lorentzian Geometry of the Heisenberg group,” Geom. Dedicata, pp. 133-140, 2006.
12. [12] K. E. Gustafson and K. M. R. Duggirala, ”Numerical Range, The Field of Values of Linear Operators and Matrices,” Springer, New York, 1997.
13. [13] M. Fiedler, ”Geometry of the numerical range of matrices,” Linear Algebra and Its Applications, vol. 37, pp. 81-96, 1981.