This is related to a question of Hoffman & Kunze, Linear Algebra (Section 6.7, #8, p. 219). The question in the text asks:
Let $T$ be a linear operator on $V$ which commutes with every projection operator on $V$. What can you say about $T$?
If $V$ is finite-dimensional, I think that $T$ is then a scalar multiple of the identity. (We know from an earlier exercise that if every subspace of a finite-dimensional vector space $V$ is invariant under $T$ then $T$ is a scalar multiple of the identity. We also know that $TE = ET \implies E(V)$ is invariant under $T$. Since we can construct a projection $E$ such that $E(V) = W$ for every subspace $W$ of $V$, we can then conclude that $T$ is a scalar multiple of the identity.)
I then have two questions: is the reasoning above for the finite-dimensional case accurate? How do we deal with the infinite-dimensional case? (It is possible the question implies that $V$ is finite-dimensional, but it does not state that, so I am uncertain.)
For background, I am self-studying linear algebra and have covered everything in Hoffman & Kunze up to this question.