Let $X, Y$ be two different Banach spaces, and let $T: X \to Y$ be a compact linear operator. Suppose the identity $I : X \to Y$ is well-defined. (For example, we could have $X = L^2([0,1])$ and $Y = L^1([0,1])$, both equipped with the Lebesgue measure).
Do most/all elementary results in spectral theory hold in that setting? For example, is it true that the spectrum of $T$ is discrete and that the eigenvalues of $T$ only accumulate at $0$? Is there a simple way to see that without reproving the result? Practically all textbooks only deal with the case where $X = Y$...