Pseudo-spectrum and the Numerical Range for Ricci Tensor on the Oscillator Group of Dimension Four

Year 2023, Volume: 20 Issue: 1, 9 - 21, 01.05.2023

Bendehiba Menad Derkaoui Rafik

Abstract

The field of functional analysis presents a very interesting part of pure mathematics, but also applied mathematics such as the theory of approximations and the resolution of operational equations, the spectra of operators, pseudo-spectrum and their numerical range which are essential techniques for researchers in several fields of science and technology. In this work, we will give the notions of the numerical range of a matrix and some properties, and study it for curvature tensor Ri j = R(∂i, ∂ j), also for Ricci tensor ρ on the oscillator group (G, ga) of dimension four and we will give examples of each matrix with the use of Matlab.

Keywords

Oscillator group, Curvature tensor, Ricci tensor, Spectrum, Pseudo-spectrum, Numerical range.

Project Number

01

Thanks

We thank all the members of this reputable journal

References

• O. Toeplitz, ”Das algebraische Analogon zu einem Staze von Feje´r,” Math. Z., vol. 2, pp. 187-197, 1918.
• F. Hausdorff, ”Der Wertvorrat einer Bilinearform,” Math. Z., vol. 3, pp. 314-316, 1919.
• A. Wintner, ”Zur Theorie der beschra¨nkten Bilinearformen,” Math. Z., vol. 30, pp. 228-282, 1929.
• M. H. Stone, ”Linear transformations in Hilbert space and their applications to analysis,” A.M.S., New York, 1932.
• R. Derkaoui and A. Smail, ”Generalized Spectrum and Numerical Range of Matrix the Lorentzian Oscillator Group of Dimension Four,” Int. J. Anal. Appl., vol. 18, no. 6, pp. 1048-1055, 2020.
• L. Reichel and L.N. Trefethen, ”Eigenvalues and pseudo-eigenvalues of Toeplitz matrices,” Linear algebra and its applications, vol. 162, pp. 153-185, 1992.
• L. Trefethen and M. Embree, ”Spectra and Pseudospectra: The Behavior of Non-Normal Matrices and Operators,” Princeton University Press, Princeton, 2005.
• K. E. Gustafson and K. M. R. Duggirala, ”Numerical Range, The Field of Values of Linear Operators and Matrices,” Springer, New York, 1997.
• G. Calvaruso and A. Zaeim, ”On the symmetries of the Lorentzian oscillator group,” Collect. Math., vol. 68, no. 51, 2017.