in general the Morse index of a critical point $p$ is the suprimum of the dimensions of sub spaces where $f''(p)$ is negative definite
but whene $f''(p)=I-T$ ($f''(p)$ is a compact perturbation of the identity) On An INFINITE Dimensional Hilbert space
The Morse index of $p$ is the dimension of the sub space of the negative eigenvectors of $f''(p)$
And my question is how the prove that the definitions are equivalent ?
Thank you