Spectral flows of dilations of Fredholm operators
  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. 
Giuseppe De Nittis, Hermann Schulz-Baldes
math-ph math.FA math.MP