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$$\mathbb Z_{2}$$-indices and factorization properties of odd symmetric Fredholm operators. (English) Zbl 1341.47014
Summary: 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 L.-K. Hua [Am. J. Math. 66, 470–488 (1944; Zbl 0063.02919)] and C. L. Siegel [ibid. 65, 1–86 (1943; Zbl 0138.31401)] for matrices. As application, it is proved that the set of odd symmetric Fredholm operators has two connected components labelled by a $$\mathbb Z_{2}$$-index given by the parity of the dimension of the kernel of $$T$$. This recovers a result of M. F. Atiyah and I. M. Singer [Publ. Math., Inst. Hautes Étud. Sci. 37, 5–26 (1969; Zbl 0194.55503)]. Two examples of $$\mathbb Z_{2}$$-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.

##### MSC:
 47A53 (Semi-) Fredholm operators; index theories 81V70 Many-body theory; quantum Hall effect 82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
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