Journal of Operator Theory

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

On the geometry of normal projections in Krein spaces

Authors: (1) Eduardo Chiumiento, (2) Alejandra Maestripieri, (3) Francisco Martínez Pería
Author institution: (1) Departamento de Matemática-FCE, Universidad Nacional de La Plata, La Plata, 1900, Argentina and Instituto Argentino de Matemática ``Alberto P. Calderón'', CONICET, Buenos Aires, 1083, Argentina
(2) Departamento de Matemática-FI, Universidad de Buenos Aires, Buenos Aires, 1063, Argentina and Instituto Argentino de Matemática ``Alberto P. Calderón'', CONICET, Buenos Aires, 1083, Argentina
(3) Departamento de Matemática-FCE, Universidad Nacional de La Plata, La Plata, 1900, Argentina and Instituto Argentino de Matemtica ``Alberto P. Calderón'', CONICET, Buenos Aires, 1083, Argentina

Summary: $ \def\h{ {\mathcal H} } \def\uj{\mathcal{U}_J} \def\e{ {\mathcal E} } \def\s{ {\mathcal S} } \def\q{ {\mathcal Q} } $ 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 $Q\mapsto QQ^\#$. 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.

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

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