Resolvent and logarithmic residues of a singular operator pencil in Hilbert spaces

The present paper considers the operator pencil A(λ)=A0+A1λ, where A0,A1≠0 are bounded linear mappings between complex Hilbert spaces and A0 is neither one-to-one nor onto. Assuming that 0 is an isolated singularity of A(λ) and that the image of A0 is closed, certain operators are defined recursively starting from A0 and A1 and they are shown to provide a characterization of the image and null space of the operators in the principal part of the resolvent and of the logarithmic residues of A(λ) at 0. The relations with the classical results based on ascent and descent in [10] are discussed. In the special case of A0 being Fredholm of index 0, the present results characterize the rank of the operators in the principal part of the resolvent, the dimension of the subspaces that define the ascent and descent, the partial multiplicities, and the algebraic multiplicity of A(λ) at 0.