An Explicit Representation of the Dominant Eigenstructure for Positive Operators on Banach Lattices

The Riesz projection and the corresponding eigenfunction of a positive operator satisfying the Doeblin condition are explicitly constructed using the partial Bell polynomials. While classical Fredholm theory requires stringent summability conditions, such as the operator being in a Schatten class to ensure the convergence of Fredholm minors, our approach utilizes the local algebraic structure induced by the Doeblin condition. We define a scalar function $D(\lambda)$ whose derivative $D'(\lambda_0)$ at the dominant eigenvalue $\lambda_0$ naturally provides the normalization constant for the projection. Consequently, an explicit functional representation of the eigenfunction is obtained as a limit of a weighted ratio of the operator's kernel, bypassing the need to solve transcendental characteristic equations.