Let $P_1,\dots, P_n$, where $n$ is at least $2$, be projections in an infinite-dimensional Banach space. Consider their convex combination
$$S=\sum_{i=1}^n t_i P_i,$$
where the $t_i$ are non-negative and sum up to $1$. (For instance, the $t_i$ might be all equal.)
Question: Is the kernel of $S$ equal to the intersection of the kernels of the $P_i$?
I can see how to prove this when the $P_i$ are of norm 1, but the case I am interested in does not assume this and in fact the $P_i$ can have norm larger than 1. Any help would be appreciated.
Another variant of the question is if there are $t_i$ as above such that the kernel of $Q$ is the intersection of the kernels of the $P_i$.
Let $$ P_1=\begin{bmatrix}0&0\\0&1\end{bmatrix},\ \ P_2=\begin{bmatrix}0&0\\-2&1\end{bmatrix}. $$ Then $$ \ker P_1=\{\begin{bmatrix} t&0\end{bmatrix}^T:\ t\in\mathbb C\},\ \ \ \ker P_2=\{\begin{bmatrix} t&2t\end{bmatrix}^T:\ t\in\mathbb C\}, $$ and so $\ker P_1\cap\ker P_2=\{0\}$.
If $S=(P_1+P_2)/2$, then $$ S=\begin{bmatrix} 0&0\\ -1&1\end{bmatrix}, $$ and so $$ \ker S=\{\begin{bmatrix} t&t\end{bmatrix}:\ t\in\mathbb C\}\ne\{0\}. $$