I encountered the result below in a paper of Claude Viterbo (Intersection de sous-variétés lagrangiennes, fonctionnelles d’action et indice des systèmes hamiltoniens, p. 379) that I was reading, and it does not have a reference. If anyone could provide me a reference, it would be very helpful.
Lemma: Let $Q^t$ be a $C^1$ family of bilinear forms defined in a Hilbert space. Let $Q^t$ be nondegenerate for $t\neq 0$ and $Q^t = U_t + C_t$, where $U_t$ is positive definite with continuous inverse and $C_t$ is compact. If $\left.\frac{d}{dt}Q^t\right|_{t=0}$ when restricted to $\ker(Q^0)$ has signature $\sigma$ and nullity $\mu$ we have \begin{equation} \sigma - \mu \leq index(Q^{-1}) - index(Q^{+1}) \leq \sigma + \mu. \end{equation}
In fact, I am trying to understand the proof of proposition 8.
Thank you very much!