See Theorem 3.4.3, page 93, of [these notes][1] for a detailed  proof of the fact that $T: H\to H$ is Fredholm if and only if  there exists  $Q:H\to H$  such that $QT-1$ is compact.  If $Q=(\lambda-T)^{-1}$ is compact then

$$Q T= K(T-\lambda)+\lambda Q=1+\lambda Q$$

so that

$$  QT-1=\lambda Q =\mbox{compact}. $$







  [1]: http://www.nd.edu/~lnicolae/Pseudo.pdf