 
Summary: Inversion of analytically perturbed linear
operators that are singular at the origin
Phil Howlett
Konstantin Avrachenkov
Charles Pearce§
Vladimir Ejov¶
January 10, 2010
Abstract
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(z)1
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 examples1
.
1 Introduction
