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REPRESENTATION THEOREMS FOR INDEFINITE QUADRATIC FORMS REVISITED

Part of: Basic linear algebra General theory of linear operators 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)
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Abstract

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

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