On the Z_2-valued index of elliptic odd symmetric operators on non-compact manifolds

We investigate elliptic operators with a symmetry that forces their index to vanish. We study the secondary index, defined modulo 2. We examine Callias-type operators with this symmetry on non-compact manifolds and establish mod 2 versions of the Gromov-Lawson relative index theorem, the Callias index theorem, and the Boutet de Monvel's index theorem for Toeplitz operators.