${\mathbb{Z}}_2$-indices and factorization properties of odd symmetric Fredholm operators

Abstract

A bounded operator $T$ on a separable, complex Hilbert space is said to be odd symmetric if $I^{\ast }{T}^{t}I=T$ 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 $T=I^{\ast }{A}^{t}IA$ 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.