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.

Final version to appear in Documenta Mathematica. Title modified. Added new result that Z_2 index is equal to the parity of the spin Chern numbers

References in corpus (2)

• Robustness of the Spin-Chern number
• Spectral flows associated to flux tubes

Cited by in corpus (12)

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• Spectral flows associated to flux tubes
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• Chern numbers as half-signature of the spectral localizer
• On the C*-algebraic approach to topological phases for insulators
• Fredholm Homotopies for Strongly-Disordered 2D Insulators
• Bulk-edge correspondence and the cobordism invariance of the index
• The cohomology invariant for class DIII topological insulators
• Skew localizer and ${\mathbb{Z}}_2$-flows for real index pairings
• Exact sequence between real and complex bivariant K theories and application to the Z2 pairing
• Topological insulators from the perspective of non-commutative geometry and index theory