  An elliptic theory is constructed for operators acting in subspaces defined
via odd pseudodifferential projections. Subspaces of this type arise as
Calderon subspaces for first order elliptic differential operators on manifolds
with boundary, or as spectral subspaces for self-adjoint elliptic differential
operators of odd order. Index formulas are obtained for operators in odd
subspaces on closed manifolds and for general boundary value problems. We prove
that the eta-invariant of operators of odd order on even-dimesional manifolds
is a dyadic rational number.
