Hoffman and Kunze Linear Algebra Exercise on projections Here is the question.

Let $T$ be a linear operator on $V$ which commutes with every projection operator on $V$. What can you say about $T$?

I want to deduce that every subspace of $V$ is $T$-invariant. I already showed that the kernel and range of every projection on $V$ is $T$-invariant, but I could not go further. Any ideas?
 A: Hint: Try to show the following.

*

*If $T$ is not diagonalizable, then there must be a subspace that fails to be $T$-invariant.

*If $T$ has two distinct eigenvalues, then there must be a subspace that fails to be $T$-invariant.

If $T$ is diagonalizable and has exactly one eigenvalue, then what can you say about $T$?
A: If you have shown that all images of projections in $V$ are $T$-invariant, then you have shown that all subspaces are $T$-invariant. Just using the weaker fact that all $1$-dimensional subspaces are $T$-invariant, all nonzero vectors are eigenvectors (since the span of a nonzero vector is $T$-invariant if and only if it is an eigenvector). That means there is but one eigenspace, and it is all of $V$. Conclude.
A: Suppose $\dim (V)\lt \infty$. Let $W$ be a subspace of $V$. Then $\exists W’$ subspace of $V$ such that $V=W\oplus W’$. By theorem 10 section 6.7, $W$ is invariant under $T$$\iff$$TE_1=E_1T$. Since $T$ commutes with every projection on $V$, we have $W$ is invariant under $T$. Thus every subspace of $V$ is invariant under $T$.
