how to compare two infinite dimensional operators

Question

Perturbation theory is crucial in quantum mechanics. I was wondering how to compare two operators in the case of two infinite-dimensional operator? Assume we have Hamiltonian $H_0$ and $V$. How formally I can show that $V \ll H_0$, such that I would expand in powers of $V$.

Answer 1

Even on infinite-dimensional operator a norm may be definied, therefore reducing the problem to a one-dimensional one, where $|V| \ll |H_0|$ is a valid assumption

Answer 2

Most likely, it is sufficient to show all the eigenvalues of $S = V H_0^{-1}$ have absolute value less than the spectral radius of whatever power series you have in mind.