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We assume that the generalized resolvent for a bounded linear operator pencilmapping one Banach space onto another has an isolated essential singularity atthe origin and is analytic on some annular region of the complex plane centredat the origin. In such cases the resolvent operator can be represented on theannulus by a convergent Laurent series and the spectral set has twocomponents---a bounded component inside the inner boundary of the annulus andan unbounded component outside the outer boundary. In this paper we prove thatthe complementary spectral separation projections on the domain space areuniquely determined by the respective generating subspaces for the associatedinfinite-length generalized Jordan chains of vectors and that the domain spaceis the direct sum of these two subspaces. We show that the images of thegenerating subspaces under the mapping defined by the pencil provide acorresponding direct sum decomposition for the range space and that this issimply the decomposition defined by the complementary spectral separationprojections on the range space. If the domain space has a Schauder basis weshow that the separated systems of fundamental equations are reduced to twosemi-infinite systems of matrix equations which can be solved recursively toobtain a basic solution and thereby determine the Laurent series coefficientsfor the resolvent operator on the given annular region.

Inversion of operator pencils on Banach space using Jordan chains when the generalized resolvent has an isolated essential singularity