Consider an iterative process of the form $x^{n+1}=A x^n + b$. When $A$ is a linear operator in $\mathbb R^n$ then the criterion of convergence is $\rho(A)<1$, where $\rho(A)$ is spectral radius of $A$.
I've seen somewhere that something similar works in general case of Banach space. So, the question is what are the known results for the case of linear operator $A$ in Banach space?
References on textbooks or papers are strongly appreciated. Thank you!
This was intended as a comment but ended up being a bit too long.
It seems like this may defy nice characterization. It is fairly clear that the sequence converges for any choice of $x^0$ and $b$ if and only if $\sum_{n=0}^\infty A^n$ converges in the strong operator topology. For a finite dimensional space, this coincides with the norm topology, and a fairly straightforward argument shows that convergence in the latter is equivalent to $\rho(A)<1$. In general, however, the strong operator topology is weaker than the norm topology. (I'm also not sure the equivalence of norm convergence of the series and $\rho(A)<1$ holds in infinite dimensions either, but I have been unable to find a proof or counterexample.)
The reason I suspect that there might not be a nice characterization of the strong convergence of $\sum_{n=0}^\infty A^n$ is that the strong operator topology can be quite unwieldy in comparison to the norm topology, or even the weak operator topology. As an example, if $\{A_n\}$ is a sequence of bounded operators on a Hilbert space $H$ and $A_n'$ is the adjoint of $A_n$, then $A_n\to0$ if and only if $A_n'\to0$ in either the norm or weak operator topologies, but this does not hold in the strong operator topology.