$\Bbb Z_{2}$-indices and factorization properties of odd symmetric Fredholm operators Article Swipe
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· 2015
· Open Access
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· DOI: https://doi.org/10.4171/dm/524
· OA: W4313901789
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.
