 
Summary: Laurent series for the inversion of perturbation operators on
Hilbert space
Phil G. Howlett \Lambda Konstantin E. Avrachenkov y
February 16, 1999
Abstract
The paper studies the perturbation operators on Hilbert spaces which depend on
complex parameter. Linear, polynomial and analytical perturbations are considered. It
is demonstrated that polynomial and analytical perturbations can be transformed to the
case of linear perturbed operators in augmented space. Then, the generalization of an
efficient algorithm of Schweitzer and Stewart can be applied to obtain a Laurent expansion
for the inverse of perturbed operators.
1 Introduction
In this paper we consider the perturbation of linear operators on Hilbert space. The pertur
bation is characterized by a complex parameter z. We study three types of perturbations:
linear, polynomial and analytic. The linear perturbation is represented by
A(z) = A+ zB; (1)
where A; B 2 L(H;K) and H;K are Hilbert spaces. The polynomial perturbation is given
by
A(z) = A 0 + zA 1 + ::: + z m Am ; (2)
where A 0 ; A 1 ; :::; Am 2 L(H;K). We refer to A(z) as a perturbed operator and to A; B as
