Z2-indices of odd symmetric Fredholm operators

A bounded operator T on a separable, complex Hilbert space is said to be odd sym-metric if I∗T tI = T where I is a real unitary satisfying I2 = −1 and T t denotes the transpose of T. The Noether index of an odd symmetric Fredholm operator vanishes, but the parity of the dimension of its kernel is shown to be a homotopy invariant that is stable under compact perturbations. The class of real skew-adjoint Fredholm operators for which Atiyah and Singer defined Z2-indices is a subset of infinite codimension within the set of odd symmetric Frehholm operators. As first example for an odd Z2-index theo-rem, a Z2-version of the Gohberg-Krein theorem is presented. An even Z2-index theorem leads to a phase label for two-dimensional topological insulators with odd time-reversal symmetry, for which non-trivial Z2-index enforces non-zero spin Chern numbers. 1 Resumé Let H be a separable, complex Hilbert space and I a real skew-adjoint unitary operator on H. Skew-adjointness of I is equivalent to I2 = −1 and implies that the spectrum of I is {−ı, ı}. Such an I exists if and only if H is even or infinite dimensional. One may assume to be I in the normal form I = ( 0 −11 0), see Proposition 6 below. This paper is about bounded linear operators T ∈ B(H) on H which are odd symmetric w.r.t. I in the sense that I ∗ T I = T ∗ ⇐ ⇒ I ∗ T t I = T, (1) where the complex conjugate T is defined in terms of complex conjugation C on H by T = CTC, and the transpose of T is T t = (T)∗. The set of bounded odd symmetric operators is denoted by B(H, I). Condition (1) looks like a quaternionic condition, but actually a quaternionic operator rather satisfies I∗TI = T and the set of quaternionic operators forms a multiplicative group, while B(H, I) does not. However, the following can easily be checked. Proposition 1 B(H, I) is a linear space an