Given:

$X$ - any Banach space
$F : X \to X$ (linear bounded and non-invertible)
$P_n$, which is projector that strongly converges to the identity operator $I$ as $n \to\infty$

Can you help me come up with a linear bounded and non-invertible operator $F$ such that the conditions are satisfied:

$P_nFP_n$ - invertible on $ImP_n$
$\|(P_nFP_n)^{-1} \| < C$ (some constant)
$P_n \to I$ strongly

If you can help at least with idea it also will be great. Thank you very much! If you think that such example doesn't exist: please can you prove it or help with idea of proving it.

I've tried a lot of things. I'm already desperate, I really hope for your help, thanks.