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Age-structured models with nonlocal diffusion arise naturally in describingthe population dynamics of biological species and the transmission dynamics ofinfectious diseases in which individuals disperse nonlocally and interact eachother and the age structure of individuals matters. In the first part of ourseries papers, we study the principal spectral theory of age-structured modelswith nonlocal diffusion of Dirichlet type. First, we provide two criteria onthe existence of principal eigenvalues by using the theory of resolventpositive operators with their perturbations. Then we define the generalizedprincipal eigenvalue and use it to investigate the influence of diffusion rateon the principal eigenvalue. In addition, we establish the strong maximumprinciple for age-structured nonlocal diffusion operators. In the second part\cite{Ducrot2022Age-structuredII} we will investigate the effects of principaleigenvalues on the global dynamics of the model with monotone nonlinearity inthe birth rate and show that the principal eigenvalue being zero is critical.

Age-structured Models with Nonlocal Diffusion of Dirichlet Type, I: Principal Spectral Theory and Limiting Properties