Assume that we have finitely many projections $P_1,\dots, P_n$ on a Banach space $X$ (take an explicit case of $X=L_p(\mu)$ if a concrete examples is better). Consider their convex combination $P=\sum_{i=1}^n \alpha_i P_i$, where $\alpha_i$ are strictly positive and sum up to 1.

Question: When is the kernel of $P$ is equal to the intersection of the kernel of the $P_i$?

The only case I know how to deal with so far is when the $P_i$ have norm one, but I'd like to know how to remove this assumption. Any help will be appreciated!