Inversion of analytically perturbed linear operators that are singular at the origin

Let H and K be Hilbert spaces and for each z∈C let A(z)∈L(H,K) be a bounded but not necessarily compact linear map with A(z) analytic on a region |z|<a. If A(0) is singular we find conditions under which A⁻¹(z) is well defined on some region 0<|z|<b by a convergent Laurent series with a finite order pole at the origin. We show that by changing to a standard Sobolev topology the method extends to closed unbounded linear operators and also that it can be used in Banach spaces where complementation of certain closed subspaces is possible. Our method is illustrated with several key examples.²