  Age-structured models with nonlocal diffusion arise naturally in describing
the population dynamics of biological species and the transmission dynamics of
infectious diseases in which individuals disperse nonlocally and interact each
other and the age structure of individuals matters. In the first part of our
series papers, we study the principal spectral theory of age-structured models
with nonlocal diffusion of Dirichlet type. First, we provide two criteria on
the existence of principal eigenvalues by using the theory of resolvent
positive operators with their perturbations. Then we define the generalized
principal eigenvalue and use it to investigate the influence of diffusion rate
on the principal eigenvalue. In addition, we establish the strong maximum
principle for age-structured nonlocal diffusion operators. In the second part
\cite{Ducrot2022Age-structuredII} we will investigate the effects of principal
eigenvalues on the global dynamics of the model with monotone nonlinearity in
the birth rate and show that the principal eigenvalue being zero is critical.
