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Zbl 0719.58030
Brousseau, V.
Espaces de Krein et index des systèmes hamiltoniens. (Krein spaces and index of Hamiltonian systems). (French)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7, No.6, 525-560 (1990). ISSN 0294-1449

A Krein space is a Hilbert space equipped with a second (hermitian) inner product called the Krein product which is symmetric but indefinite, such that the operator expressing it with respect to the original one is bounded with bounded inverse. If a normal operator A is moreover either symmetric, or antisymmetric, or orthogonal, with respect to the Krein product, let $\lambda$ be an eigenvalue of finite multiplicity of A which is isolated in the spectrum and is real, or purely imaginary, or of modulus 1, respectively. Then the Krein product induces on the eigenspace $E\sb{\lambda}$ a non degenerate inner product whose signature (number of positive minus number of negative squares) is called the signature of $\lambda$. The sum over all these eigenvalues, weighted with their signatures, is called the Krein trace of A; the product over all these eigenvalues $\lambda$ of $\lambda\sp{sgk \lambda}$ is called the Krein determinant of A. The Krein trace is a continuous function of A, homogeneous of degree 1, but not linear. It is shown explicitly how to compute these from the Jordan form of a matrix. It is shown that the Morse-Ekeland index [{\it I. Ekeland}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 19-78 (1984; Zbl 0537.58018)] and the index of {\it C. Conley} and {\it E. Zehnder} [Commun. Pure Appl. Math. 37, 207-253 (1984; Zbl 0559.58019)] admit common generalizations in terms of the Krein trace.
[P.Michor (Wien)]

MSC 2000:: *37A30 Ergodic theorems, spectral theory, Markov operators
37J99 Finite-dimensional Hamiltonian etc. systems
47A75 Eigenvalue problems (linear operators)
46C20 Spaces with indefinite inner product
58J20 Index theory and related fixed point theorems

Keywords: Hamiltonian systems; Hilbert product; resolvant; symplectic operator; Krein product; normal operator; Krein determinant; Krein trace; index

Citations: Zbl 0537.58018; Zbl 0559.58019

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