We propose a general framework for a sufficient condition of the existence of the principal eigenpair to the eigenvalue problem L[u]+H[u] = λu, (λ, u) ∈ R×X, where L : X → X is a positive bounded linear operator and H : D(H) ⊂ X → X a closed, possibly unbounded linear operator in the Banach space (X, ∥ • ∥). Criteria for (i) the existence of a principal eigenpair, (ii) Asynchronous Exponential Growth (AEG) and, (iii) continuity result of the spectral bound are given, without necessarily specifying forms of operators for the above problem. The criteria are then applied to generalize some results in the existing literature in context of nonlocal dispersal operators. Our results are applied to a model of a chemostat and a model of a spatial evolution of a man-environment disease.