Z_2 indices and factorization properties of odd symmetric Fredholm operators

TL;DR

This paper explores the properties of odd symmetric Fredholm operators, establishing a factorization theorem, analyzing their topological classification via a Z2-index, and illustrating applications in index theory and topological insulators.

Contribution

It generalizes factorization results for odd symmetric operators and introduces a Z2-index classification for Fredholm operators with applications in topology and physics.

Findings

Operators can be factorized as T=I^*A^tIA.

The set of odd symmetric Fredholm operators has two connected components.

Applications include Z2-index theorems and topological insulators.

Abstract

A bounded operator $T$ on a separable, complex Hilbert space is said to be odd symmetric if $I^*T^tI=T$ 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 $T=I^*A^tIA$ 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.