Geodesic
$\Bbb Z_{2}$-indices and factorization properties of odd symmetric Fredholm operators
Documenta mathematica, Tome 20 (2015), pp. 1481-1500.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Classification : 47A53, 81V70, 82D30 
@article{DOCMA_2015__20__a0,
     author = {Schulz-Baldes, Hermann},
     title = {$\Bbb Z_{2}$-indices and factorization properties of odd symmetric {Fredholm} operators},
     journal = {Documenta mathematica},
     pages = {1481--1500},
     publisher = {mathdoc},
     volume = {20},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2015__20__a0/}
}
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Schulz-Baldes, Hermann. $\Bbb Z_{2}$-indices and factorization properties of odd symmetric Fredholm operators. Documenta mathematica, Tome 20 (2015), pp. 1481-1500. http://geodesic.mathdoc.fr/item/DOCMA_2015__20__a0/