an:06572187
Zbl 1341.47014
Schulz-Baldes, Hermann
\(\mathbb Z_{2}\)-indices and factorization properties of odd symmetric Fredholm operators
EN
Doc. Math. 20, 1481-1500 (2015).
00342805
2015
j
47A53 81V70 82D30
\(\mathbb Z_{2}\)-valued index theorems; Noether-Gohberg-Krein theorem with symmetries; topological insulators
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 \textit{L.-K. Hua} [Am. J. Math. 66, 470--488 (1944; Zbl 0063.02919)] and \textit{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 \textit{M. F. Atiyah} and \textit{I. M. Singer} [Publ. Math., Inst. Hautes ??tud. 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.
Zbl 0063.07003; Zbl 0063.02919; Zbl 0138.31401; Zbl 0194.55503