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 $X = Y$...



Spectrum of compact operator between different Banach spaces
By "$I$ is well-defined", I presume you mean you have a continuous injection $\iota$ of $X$ into $Y$. If $X$ and $Y$ are not isomorphic it will not be a bijection, because there is no continuous linear bijection from $X$ to $Y$, and the same applies to $T - \lambda \iota$. Thus $T - \lambda \iota$ will never be invertible, and the "spectrum" of $T$ will be all of $\mathbb C$. 


