Z2-Indices and Factorization Properties of Odd Symmetric Fredholm Operators

A bounded operator T on a separable, complex Hilbert space is said to be odd symmetric if IT I = T where I is a real unitary satisfying I = −1 and T t denotes the transpose of T . It is proved that such an operator can always be factorized as T = IAIA 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 Z2index given by the parity of the dimension of the kernel of T . This recovers a result of Atiyah and Singer. Two examples of Z2-valued index theorems are provided, one being a version of the NoetherGohberg-Krein theorem with symmetries and the other an application to topological insulators. 2010 Mathematics Subject Classification: 47A53, 81V70, 82D30