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REPRESENTATION THEOREMS FOR INDEFINITE QUADRATIC FORMS REVISITED

Part of: General theory of linear operators Basic linear algebra Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  12 June 2012

Luka Grubišić ,
Vadim Kostrykin ,
Konstantin A. Makarov  and
Krešimir Veselić
Luka Grubišić
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia (email: luka.grubisic@math.hr)
Vadim Kostrykin
Affiliation:
Institut für Mathematik FB 08, Johannes Gutenberg-Universität Mainz, Staudinger Weg 9, D-55099 Mainz, Germany (email: kostrykin@mathematik.uni-mainz.de)
Konstantin A. Makarov
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A. (email: makarovk@missouri.edu)
Krešimir Veselić
Affiliation:
Fakultät für Mathematik und Informatik, Fernuniversität Hagen, Postfach 940, D-58084 Hagen, Germany (email: kresimir.veselic@fernuni-hagen.de)

Abstract

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The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New and straightforward proofs of these theorems are given. A number of necessary and sufficient conditions for the second representation theorem to hold are obtained. A new simple and explicit example of a self-adjoint operator for which the second representation theorem fails to hold is also provided.

MSC classification

Secondary: 47A07: Forms (bilinear, sesquilinear, multilinear) 47A55: Perturbation theory 15A63: Quadratic and bilinear forms, inner products 46C20: Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.)

Information

Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 169 - 189
Copyright
Copyright © University College London 2012

References

[1]Azizov, T. Ya. and Iokhvidov, I. S., Linear Operators in Spaces with an Indefinite Metric, Wiley (Chichester, 1989).Google Scholar
[2]Birman, M. Sh. and Solomyak, M. Z., Spectral Theory of Selfadjoint Operators in Hilbert Space, D. Reidel Publishing Co. (Dordrecht, 1987).CrossRefGoogle Scholar
[3]Bognár, J., Indefinite Inner Product Space (Ergebnisse der Mathematik und ihrer Grenzgebiete 78), Springer (Berlin, 1974).CrossRefGoogle Scholar
[4]van Casteren, J. A., Operators similar to unitary or selfadjoint ones. Pacific J. Math. 104 (1983), 241–255.CrossRefGoogle Scholar
[5]Ćurgus, B., On the regularity of the critical point infinity of definitizable operators. Integral Equations Operator Theory 8 (1985), 462–488.CrossRefGoogle Scholar
[6]Edmunds, D. E. and Evans, W. D., Spectral Theory and Differential Operators (Oxford Mathematical Monographs), Clarendon Press (Oxford, 1989).Google Scholar
[7]Esteban, M. J. and Loss, M., Self-adjointness via partial Hardy-like inequalities. In Mathematical Results in Quantum Mechanics, (eds Beltita, I., Nenciu, G. and Purice, R.), World Scientific (Hackensack, NJ, 2008), 41–47.CrossRefGoogle Scholar
[8]Fleige, A., Spectral Theory of Indefinite Krein–Feller Differential Operators (Mathematical Research 98), Akademie (Berlin, 1996).Google Scholar
[9]Fleige, A., A counterexample to completeness properties for indefinite Sturm–Liouville problems. Math. Nachr. 190 (1998), 123–128.CrossRefGoogle Scholar
[10]Fleige, A., Non-semibounded sesquilinear forms and left-indefinite Sturm–Liouville problems. Integral Equations Operator Theory 33 (1999), 20–33.CrossRefGoogle Scholar
[11]Fleige, A., Hassi, S. and de Snoo, H., A Krein space approach to representation theorems and generalized Friedrichs extensions. Act. Sci. Math. (Szeged) 66 (2000), 633–650.Google Scholar
[12]Fleige, A., Hassi, S. and de Snoo, H., Generalized Friedrichs extensions associated with interface conditions for Sturm–Liouville operators. In Operator Theory and Indefinite Inner Product Spaces (Operator Theory: Advances and Applications 163) (eds Langer, M., Luger, A. and Woracek, H.), Birkhäuser (Basel, 2006), 135–145.Google Scholar
[13]Fleige, A., Hassi, S., de Snoo, H. and Winkler, H., Sesquilinear forms corresponding to a non-semibounded Sturm–Liouville operator. Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), 291–318.CrossRefGoogle Scholar
[14]Gesztesy, F., Malamud, M., Mitrea, M. and Naboko, S., Generalized polar decompositions for closed operators in Hilbert spaces and some applications. Integral Equations Operator Theory 64 (2009), 83–113.CrossRefGoogle Scholar
[15]Heinz, E., Beiträge zur Störungstheorie der Spektralzerlegung. Math. Ann. 123 (1951), 415–438.CrossRefGoogle Scholar
[16]Kato, T., Perturbation Theory for Linear Operators, Springer (Berlin, 1966).Google Scholar
[17]Konstantinov, A. and Mennicken, R., On the Friedrichs extension of some block operator matrices. Integral Equations Operator Theory 42 (2002), 472–481.CrossRefGoogle Scholar
[18]Kostrykin, V., Makarov, K. A. and Motovilov, A. K., A generalization of the tan 2Θ theorem. In Current Trends in Operator Theory and Its Applications (Operator Theory: Advances and Applications 149) (eds Ball, J. A., Klaus, M., Helton, J. W. and Rodman, L.), Birkhäuser (Basel, 2004), 349–372.CrossRefGoogle Scholar
[19]Kostrykin, V., Makarov, K. A. and Motovilov, A. K., Perturbation of spectra and spectral subspaces. Trans. Amer. Math. Soc. 359 (2007), 77–89.CrossRefGoogle Scholar
[20]Krein, S. G., Petunin, Ju. I. and Semenov, E. M., Interpolation of Linear Operators (Translations of Mathematical Monographs 54), American Mathematical Society (Providence, RI, 1982).Google Scholar
[21]Langer, H., Spectral functions of definitizable operators in Krein spaces. In Functional Analysis (Lecture Notes in Mathematics 948) (eds H., D. and Kurepa, Butković), Springer (Berlin, 1982), 1–46.Google Scholar
[22]McIntosh, A., Bilinear forms in Hilbert space. J. Math. Mech. 19 (1970), 1027–1045.Google Scholar
[23]McIntosh, A., Hermitian bilinear forms which are not semibounded. Bull. Amer. Math. Soc. 76 (1970), 732–737.CrossRefGoogle Scholar
[24]Nenciu, G., Self-adjointness and invariance of the essential spectrum for Dirac operators defined as quadratic forms. Comm. Math. Phys. 48 (1976), 235–247.CrossRefGoogle Scholar
[25]Veselić, K., Perturbation theory for the eigenvalues of factorised symmetric matrices. Linear Algebra Appl. 309 (2000), 85–102.CrossRefGoogle Scholar
[26]Veselić, K., Spectral perturbation bounds for sefadjoint operators. I. Oper. Matrices 2 (2008), 307–339.CrossRefGoogle Scholar
[27]Volkmer, H., Sturm–Liouville problems with indefinite weights and Everitt’s inequality. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 1097–1112.CrossRefGoogle Scholar