Suppose we have a bounded linear operator $A = A(\gamma):H\_1\to H\_2$ where $H\_1$ and $H\_2$ are Hilbert spaces and $\gamma>0$ is some parameter, and we are interested in the solution to
$$
(I-A)x = y.
$$
If $\|A\|<1$ we can use a Neumann series expansion and get a series representation:
\begin{align}
x & = (I-A)^{-1} y \\
& = \sum\_{j=0}^\infty A^j y
\end{align}


Now, suppose that when $\gamma$ becomes small enough $\|A(\gamma)\|$ becomes greater than $1$ and the Neumann series won't converge; thus we can't get a series representation for $x$.


An example of this situation can be found in problems featuring scattering of waves among disjoint objects. If the solution is represented as a Neumann series, it can fail to converge if frequencies become high or if distances between objects become small.


Are any techniques that can used in such a situation to 'get around' the non-convergence of a Neumann series and obtain a series representation for $x$?



Are there any techniques that can be used in the case when a Neumann series doesn't converge?
If the spectrum of $A$ is contained in a disk $\{z: |z - a| \le r\}$ where $|1-a| > r$, then the series $\sum\_{n=0}^\infty (1-a)^{-1-n} (A - a I)^n$ converges to $(I-A)^{-1}$.


