Eigenvalues and Eigenvectors of continous variable function

Question

Basically I am given a function $P_{l,k}$ where P is continuous and hermitian in $l$ and $k$ i.e. $P_{l,k}^*=P_{k,l}$. What is the general procedure to find its eigenvalues and eigenfunctions? There must be some way of generalizing the meaning of a determinant to continuous functions, but I can't find anything in the literature.