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
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