On the geometry of normal projections in Krein spaces

Abstract

Let H be a Krein space with fundamental symmetry J. Along this paper, the geometric structure of the set of J-normal projections Q is studied. The group of J-unitary operators UJ naturally acts on Q. Each orbit of this action turns out to be an analytic homogeneous space of UJ, and a connected component of Q. The relationship between Q and the set E of J-selfadjoint projections is analized: both sets are analytic submanifolds of L(H) and there is a natural real analytic submersion from Q onto E, namely Q↦QQ#. The range of a J-normal projection is always a pseudo-regular subspace. Then, for a fixed pseudo-regular subspace S, it is proved that the set of J-normal projections onto S is a covering space of the subset of J-normal projections onto S with fixed regular part.