Critical group and morse Lemma

Question

I have this theorem with a part of it's prove

I have two questions:

1) what is the spectral decomposition of A ?

2)How to see that $B_{\varepsilon}\cap f_0 = \lbrace x\in H,||x||\leq \varepsilon , ||y_+||\leq ||y_-||,y_{±}=(±AP_{±})^{\frac12}x\rbrace$

Please,

Thank you .