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ALGEBRAIC ASPECTS OF SPECTRAL THEORY

Part of: Individual linear operators as elements of algebraic systems General theory of linear operators Basic linear algebra

Published online by Cambridge University Press:  21 December 2010

E. B. Davies
E. B. Davies*
Affiliation:
Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, U.K. (email: [email protected])
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Abstract

We describe some aspects of spectral theory that involve algebraic considerations but need no analysis. Some of the important applications of the results are to the algebra of n×n matrices with entries that are polynomials or more general analytic functions.

MSC classification

Secondary: 47C05: Operators in algebras 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones) 15A22: Matrix pencils
Type
Research Article
Information
Mathematika , Volume 57 , Issue 1 , January 2011 , pp. 63 - 88
Copyright
Copyright © University College London 2011

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References

[1]Atiyah, M. F. and Macdonald, I. G., Introduction to Commutative Algebra, Addison-Wesley (Reading, MA, 1969).Google Scholar
[2]Ball, J. A. and Rakowski, M., Minimal McMillan degree rational matrix functions with prescribed local zero-pole structure. Linear Algebra Appl. 137–138 (1990), 325–349.Google Scholar
[3]Bhatia, R. and Rosenthal, P., How and why to solve the operator equation AX−XB=Y. Bull. Lond. Math. Soc. 29 (1997), 1–21.CrossRefGoogle Scholar
[4]Bourbaki, N., Éléments de mathématiques, fasc. 32, Théories spectrales, Hermann (Paris, 1967), Ch. 1, 2.Google Scholar
[5]Carter, R. W., Simple Groups of Lie Type, John Wiley and Sons (London, 1989).Google Scholar
[6]Coppel, W. A., Matrices of rational functions. Bull. Aust. Math. Soc. 11 (1974), 89–113.CrossRefGoogle Scholar
[7]Dales, G., Banach Algebras and Automatic Continuity (London Mathematical Society Monographs 2), Clarendon Press (Oxford, 2000).Google Scholar
[8]Davies, E. B., Linear Operators and their Spectra, Cambridge University Press (Cambridge, 2007).CrossRefGoogle Scholar
[9]Davies, E. B., Decomposing the essential spectrum. J. Funct. Anal. 257 (2009), 506–536.Google Scholar
[10]Dedier, J.-P. and Tisseur, F., Perturbation theory for homogeneous polynomial eigenvalue problems. Linear Algebra Appl. 358 (2003), 71–94.Google Scholar
[11]Duffin, R. J. and Hazony, D., The degree of a rational matrix function. SIAM J. Appl. Math. 11 (1963), 645–658.CrossRefGoogle Scholar
[12]Edmunds, D. E. and Evans, W. D., Spectral Theory and Differential Operators, Oxford University Press (Oxford, 1987).Google Scholar
[13]Flanders, H. and Wimmer, H. K., On the matrix equations AX−XB=G and AX−Y B=C. SIAM J. Appl. Math. 32 (1977), 707–710.CrossRefGoogle Scholar
[14]Georgescu, V. and Iftimovici, A., C *-algebras of quantum Hamiltonians. In Operator Algebras and Mathematical Physics (Proc. Conf. Operator Algebras and Mathematical Physics, Constanta 2001, Ed. Theta) (eds J.-M. Combes, J. Cuntz, G. A. Elliot, G. Nenciu, H. Siedentop and S. Stratila) (2003) 123–167.Google Scholar
[15]Georgescu, V. and Iftimovici, A., Localizations at infinity and essential spectrum of quantum Hamiltonians: I. General theory. Rev. Math. Phys. 18 (2006), 417–483.Google Scholar
[16]Gohberg, I., Lancaster, P. and Rodman, L., Matrix Polynomials, Academic Press (New York, 1982).Google Scholar
[17]Lancaster, P. and Rodman, L., Algebraic Riccati Equations, Oxford University Press (Oxford, 1995).Google Scholar
[18]Marcus, A. S. and Merentsa, I. V., On some properties of λ-matrices. Math. Issled, Kishinev 3 (1975), 207–213 (in Russian).Google Scholar
[19]Nevanlinna, R. and Paatero, V., Introduction to Complex Analysis, Addison-Wesley (Reading, MA, 1964).Google Scholar
[20]Peller, V. V., Hankel Operators and their Applications, Springer (New York, 2003).Google Scholar
[21]Pressley, A. and Segal, G., Loop Groups (Oxford Mathematical Monographs), Oxford University Press (Oxford, 1986).Google Scholar
[22]Rosenblum, M., On the operator equation BX−XA=Q. Duke Math. J. 23 (1956), 263–269.CrossRefGoogle Scholar
[23]Rosenbrock, H. H., State-Space and Multivariable Theory, Thomas Nelson and Sons (London, 1970).Google Scholar
[24]Roth, W. E., The equation AX−Y B=C and AX−XB=C in matrices. Proc. Amer. Math. Soc. 3 (1952), 392–396.Google Scholar
[25]Saks, S. and Zygmund, A., Analytic Functions. Transl. E. J. Scott, Monografie Matematyczne, Tom 28, (Warsaw, 1952).Google Scholar
[26]Sz.-Nagy, B. and Foiaş, C., Harmonic Analysis of Operators on Hilbert Space, North-Holland (Amsterdam, 1970).Google Scholar
[27]Tao, T., 254A, Notes 5: free probability,http://terrytao.wordpress.com/2010/02/10/245a-notes-5-free-probability/.Google Scholar
[28]Voiculescu, D. V., Dykema, K. J. and Nica, A., Free random variables. In A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups (CRM Monograph Series 1), American Mathematical Society (Providence, RI, 1992).Google Scholar
[29]Whitney, H., Complex Analytic Varieties, Addison-Wesley (Reading, MA, 1972).Google Scholar
[30]Zhou, K., Doyle, J. C. and Glover, K., Robust and Optimal Control, Prentice Hall (Upper Saddle River, NJ, 1996).Google Scholar