Some results on S-numerical range of operator matrices
DOI:
https://doi.org/10.21271/ZJPAS.32.3.7Keywords:
S-numerical range; projection method; Schro ̈dinger operator; Hain-Lu ̈st operator; Stokes operator.Abstract
A linear operator on a Hilbert space may be approximated with finite matrices by choosing an orthonormal basis of the Hilbert space. In this paper, we found an approximation of the S-numerical range of bounded and unbounded operator matrices by variation methods. Applications to Hain-L st operator and Stokes operator are given.
References
ADAMJAN, V. M. & LANGER, H. 1995. Spectral properties of a class of rational operator valued functions. Journal of Operator Theory, 259-277.
BEBIANO, N., LEMOS, R., DA PROVIDENCIA, J. & SOARES, G. 2004. On generalized numerical ranges of operators on an indefinite inner product space. Linear and Multilinear Algebra, 52, 203-233.
BEBIANO, N., LEMOS, R., DA PROVIDÊNCIA, J. & SOARES, G. 2005. On the geometry of numerical ranges in spaces with an indefinite inner product. Linear algebra and its applications, 399, 17-34.
BEBIANO, N. & PROVIDÊNCIA, J. O. D. 1998. Numerical ranges in physics. Linear and Multilinear Algebra, 43, 327-337.
BEBIANO, N., PROVIDIA, J. D., NATA, A. & SOARES, G. 2008. Krein Spaces Numerical Ranges and their Computer Generation, Electron. J. Linear Algebra,.
GUSTAFSON, K. E. & RAO, D. K. 1997. Numerical range. Numerical Range. Springer.
HAIN, K. & LUST, R. 1958. Zur Stabilität zylindersymmetrischer Plasmakonfigurationen mit Volumenströmen. Zeitschrift für Naturforschung A, 13, 936-940.
HALMOS, P. R. 2012. A Hilbert space problem book, Springer Science & Business Media.
HAUSDORFF, F. 1919. Der wertvorrat einer bilinearform. Mathematische Zeitschrift, 3, 314-316.
KATO, T. 2013. Perturbation theory for linear operators, Springer Science & Business Media.
LANGER, H., MENNICKEN, R. & MÖLLER, M. 1990. A second order differential operator depending nonlinearly on the eigenvalue parameter. Oper. Theory Adv. Appl, 48, 319-332.
LANGER, H. & TRETTER, C. 1998. Spectral decomposition of some nonselfadjoint block operator matrices. Journal of Operator Theory, 339-359.
LI, C.-K., TSING, N.-K. & UHLIG, F. 1996. Numerical ranges of an operator on an indefinite inner product space. Electronic Journal of Linear Algebra.
MURNAGHAN, F. D. 1932. On the field of values of a square matrix. Proceedings of the National Academy of Sciences of the United States of America, 18, 246.
NAKAZATO, H., BEBIANO, N. & DA PROVIDÊNCIA, J. 2011. THE NUMERICAL RANGE OF LINEAR OPERATORS ON THE 2-DIMENTIONAL KREIN SPACE. ELECTRONIC JOURNAL OF LINEAR ALGEBRA, 22, 430-442.
R.A.HORN & C.R.JOHNSON 1991. "Topics in matrix analysis", Cambridge University Press, New York.
TAHIRI, F. E. 2015. Numerical Ranges of Linear Pencils. PhD thesis, University of Coimbra.
TOEPLITZ, O. 1918. Das algebraische analogon zu einem salze you fient, Math. Z. Vol.2, (1918),187-197.
BEBIANO, N., PROVIDIA, J. D., NATA, A. & SOARES, G. 2008. Krein Spaces Numerical Ranges and theirComputer Generation, Electron. J. Linear Algebra,.
GUSTAFSON, K. E. & RAO, D. K. 1997. Numerical range. Numerical Range. Springer.
HALMOS, P. R. 2012. A Hilbert space problem book, Springer Science & Business Media.
LI, C.-K., TSING, N.-K. & UHLIG, F. 1996. Numerical ranges of an operator on an indefinite inner product space. Electronic Journal of Linear Algebra.
MUHAMMAD, A. M. S. 2005a. Elliptical range of n-tuple operators on a complex Hilbert space, Zanko Journal for Pure and Applied Science.
MUHAMMAD, A. M. S. 2005b. Line segments of the boundary of numerical range, Zanko Journal for Pure and Applied Science.
N.BEBIANO, LEMOS, R., PROVIDENCIA, J. D. & SOARES, G. 2004. On generalized numerical ranges of operators on an indenite inner product space, Linear and Multilinear Algebra, 52:203233. Mathematische Annalen.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2020 Berivan Faris Azeez , Ahmed Muhammad
This work is licensed under a Creative Commons Attribution 4.0 International License.