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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 √Čtud. 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.

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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