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 ( 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 examples.22This paper is based on preliminary work in [P.G. Howlett, K.E. Avrachenkov, Laurent series for the inversion of perturbed lionear operators on Hilbert space, in: A. Rubinov (Ed.), Progress in Optimisation III, Contributions from Australasia, Kluwer, 2001, pp. 325–342; P.G. Howlett, V. Ejov, K.E. Avrachenkov, Inversion of perturbed linear operators that are singular at the origin, in: Proceedings of 42nd IEEE Conference on Decision and Control, Maui, Hawai, December 2003, pp. 5628–5631 (on compact disc)].