# $Z_{2}$-indices and factorization properties of odd symmetric Fredholm operators

### Hermann Schulz-Baldes

Department Mathematik Universität Erlangen-Nürnberg Germany

## Abstract

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.

## Cite this article

Hermann Schulz-Baldes, $Z_{2}$-indices and factorization properties of odd symmetric Fredholm operators. Doc. Math. 20 (2015), pp. 1481–1500

DOI 10.4171/DM/524