@mathreadler $\text{null}(\cdots)$ and $\text{range}(\cdots)$ are subspaces; you may be conflating them with nullity and rank which are integers

For any operator, if $T$ is one-to-one and $\lambda$ is not an eigenvalue, then $\mathrm{null}(T-\lambda I)$ is trivial, and by the rank-nullity Theorem, $T-\lambda I$ is onto, so you get the sum you want. Even when $T$ is not diagonalizable.

@angryavian Yes, this is for all $\lambda \in \mathbb{C}$.

@N.S.: No, because you need the range and nullspace to intersect trivially for the right hand side to be a direct sum.

@ArturoMagidin Yep. This is probably what I meant, but did not manage to write.

If you are working over $\mathbb{C}$ and the equality is required for all complex numbers $\lambda$ (or at least, all eigenvalues $\lambda$), the answer is going to be “yes” thanks to the theory of generalized eigenvectors. In order for the nullspace and range to intersect trivially, all generalized eigenvectors will have to actually be eigenvectors, and that is known to be necessary and sufficient for diagonalizability.

But I’ll wager that at this point you don’t know about generalized eigenvectors, in which case this is going to be hard to prove, I think.

@ArturoMagidin My text doesn't go into generalized eigenvectors until much later, and my book solution uses answers to previous questions to solve this, which I don't think is an appropriate way for me to answer this. I was thinking there must be some way to use theorems of direct sums or eigenspaces to solve this, but haven't made progress.

@Chris: Sorry; I don’t know your book (you never say which book you are using), nor did you say what you knew. I thought this was something you had come up with or run across, not a specific problem from a specific book that you are expected to solve with specific tools.

@chris: In other words: if you want an answer restricted to certain tools, you need to tells us that restriction!

@Chris: Put it in the question! Don’t force people to wade through a long comment thread to get the relevant information.

Just updated the question, sorry

I can’t look at my copy of Axler until Wednesday (campus closes for Carnival, and I’m locked out until then); I’ll take a look then to get the context and see if I can think of a way to get at it.

@ArturoMagidin No worries if you can't. I can try to discuss it with my professor this week as well.

Is this an exercise in the book? Or an assignment that was made? In either case, which section of the book does it correspond to? (If it's an exercise, what chapter and number?)