I learned about a theorem which states:
Let $A \in \mathbb C^{n \times n}$. The Neumann series converges if
and only if it holds for the spectral radius that $r(A) < 1$. In this case, $I − A$ is invertible and we have
$$ \sum_{n = 0}^\infty A^n = (I – A)^{-1}.$$
I could show one implication of this theorem in a Banach space setting:
Let $X$ a Banach space and $T: X \to X$ a compact operator. If the Neumann series convergences it holds that $r(T) < 1$.
But I have no idea how to show the other implication in this setting.
I wondered if one can make such a theorem work for operators on Banach spaces or Hilbert spaces? Is there a analogous theorem for a suitable infinite dimensional setting (maybe compact operators)? If there is one, I would really enjoy some literature advice, because I couldn't find anything related to that using google. Thanks in advance 🙂
Best Answer
Recall that the spectral radius satisfies
$$r(T) = \lim_{n\to \infty}\, \lVert T^n\rVert^{1/n}.$$
Then choose any $q \in \bigl(r(T),1\bigr)$. For all large enough $n$, we have $\lVert T^n\rVert < q^n$, and we see that the Neumann series is (absolutely) convergent when $r(T) < 1$.