Geodesic
$\Bbb Z_{2}$-indices and factorization properties of odd symmetric Fredholm operators
Documenta mathematica, Tome 20 (2015), pp. 1481-1500
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A bounded operator T on a separable, complex Hilbert space is said to be odd symmetric if I∗TtI=T where I is a real unitary satisfying I2=−1 and T^t denotes the transpose of T. It is proved that such an operator can always be factorized as T=I∗AtIA 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.
DOI : 10.4171/dm/524
Classification : 47A53, 81V70, 82D30 
@article{10_4171_dm_524,
     author = {Hermann Schulz-Baldes},
     title = {$\Bbb Z_{2}$-indices and factorization properties of odd symmetric {Fredholm} operators},
     journal = {Documenta mathematica},
     pages = {1481--1500},
     year = {2015},
     volume = {20},
     doi = {10.4171/dm/524},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/524/}
}
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Hermann Schulz-Baldes. $\Bbb Z_{2}$-indices and factorization properties of odd symmetric Fredholm operators. Documenta mathematica, Tome 20 (2015), pp. 1481-1500. doi: 10.4171/dm/524

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