Suppose that $V$ is a $\mathbb C$-vector space. I'm eventually interested in the infinite-dimensional case, but let's say for now that it's finite dimensional. Suppose that $\mathscr S$ is a collection of rank-1 projectors on $V$, not necessarily commuting or respecting any interesting inner product, such that $\sum\_{p \in \mathscr S} pV = V$; and that $T$ is an endomorphism of $V$ such that $\operatorname{tr} T \ne 0$. Is there necessarily some $p \in \mathscr S$ so that $p T p \ne 0$? I think of this loosely as: is there necessarily some line in a collection of lines that witnesses a non-0 trace?


(This question arises for me while trying to generalise some old work of Moy and Prasad on minimal K-types. They need only to deduce from the fact that $\operatorname{tr} T \ne 0$ that there is some $p \in \mathscr S$ with $T p \ne 0$, which is clear; but I seem to need the stronger version above.)



Traces and projectors
Counterexample:


$V$ is $2$-dimensional with basis $\left(e,f\right)$. The projectors are projecting on $\mathbb C e$ and $\mathbb C f$, each with kernel $e+f$. The map $T$ projects on $\mathbb C \left(e+f\right)$.


