  Given an essentially unitary contraction and an arbitrary unitary dilation of
it, there is a naturally associated spectral flow which is shown to be equal to
the index of the operator. This purely operator theoretic result is interpreted
in terms of the $K$-theory of an associated mapping cone. It is then extended
to connect $Z_2$ indices of odd symmetric Fredholm operators to a $Z_2$-valued
spectral flow.
