# Journal of Operator Theory

Volume 74, Issue 1, Summer 2015  pp. 75-99.

On the geometry of normal projections in Krein spaces

Authors:  Eduardo Chiumiento (1), Alejandra Maestripieri (2), and Francisco Martinez Peria (2)
Author institution:(1) Departamento de Matem\'atica--FCE, Universidad Nacional de La Plata, La Plata, 1900, Argentina and Instituto Argentino de Matem\'atica Alberto P. Calder\'on, CONICET, Buenos Aires, 1083, Argentina
(2) Departamento de Matem\'atica--FI, Universidad de Buenos Aires, Buenos Aires, 1063, Argentina and Instituto Argen\-tino de Matem\'atica Alberto P. Calder\'on, CONICET, Buenos Aires, 1083, Argentina
(3) Departamento de Matem\'atica--FCE, Universidad Nacional de La Plata, La Plata, 1900, Argentina and Instituto Argentino de Matem\'atica Alberto P. Calder\'on, CONICET, Buenos Aires, 1083, Argentina

Summary: 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 $\q$. 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 $Q\mapsto QQ^\sharp$. 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.

DOI: http://dx.doi.org/10.7900/jot.2014may06.2035
Keywords: Krein space, normal operator, projection, submanifold