The fundamental equations for the generalized resolvent of an elementary pencil in a unital Banach algebra

We show that the generalized resolvent of a linear pencil in a unital Banach algebra over the field of complex numbers is analytic on an open annular region of the complex plane if and only if the coefficients of the Laurent series expansion satisfy a system of left and right fundamental equations and are geometrically bounded. Our analysis includes the case where the resolvent has an isolated essential singularity at the origin. We find a closed form for the resolvent and use the fundamental equations to establish key spectral separation properties when the resolvent has only a finite number of isolated singularities. We show that our results can be used to solve an infinite system of ordinary differential equations and to solve the generalized Sylvester equation. We also show that our results can be extended to polynomial pencils.