Z_{2}-Indices and Factorization Properties
of Odd Symmetric Fredholm Operators

A bounded operator $T$ on a separable, complex Hilbert space is said to
be odd symmetric if $I^*T^{t}I=T$ where $I$ is a real unitary satisfying
$I^{2}=-1$ and $T^{t}$ denotes the transpose of $T$. It is proved that such
an operator can always be factorized as $T=I^*A^{t}IA$ with some operator
$A$. This generalizes a result of Hua and Siegel for matrices. As application
it is proved that the set of odd symmetric Fredholm operators has two
connected components labelled by a $Z_{2}$-index given by the parity of
the dimension of the kernel of $T$. This recovers a result of Atiyah and
Singer. Two examples of $Z_{2}$-valued index theorems are provided, one
being a version of the Noether-Gohberg-Krein theorem with symmetries and
the other an application to topological insulators.

2010 Mathematics Subject Classification: 47A53, 81V70, 82D30

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