On the geometry of normal projections in Krein spaces

TL;DR

This paper explores the geometric structure of $J$-normal projections in Krein spaces, revealing their manifold properties, group actions, and relationships with $J$-selfadjoint projections, advancing the understanding of their geometric and algebraic features.

Contribution

It characterizes the geometric and manifold structure of $J$-normal projections and their relation to $J$-selfadjoint projections in Krein spaces, including orbit and submersion properties.

Findings

Orbits of $J$-unitary group actions are analytic homogeneous spaces.

There is a natural real analytic submersion from $J$-normal projections to $J$-selfadjoint projections.

The set of $J$-normal projections onto a fixed pseudo-regular subspace forms a covering space.

Abstract

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