  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.
