Second-Kind Linear Volterra Integral Equations with Noncompact Operators

In this article, we consider second-kind linear Volterra integral equations (VIEs) with noncompact operators, that is, μu = f + V ϕ u, where with the core ϕ ∈L 1(0, 1). We present some different properties of noncompact operators V ϕ from compact operators, such as eigenvalues, eigenfunctions, null spaces, and ranges. In many applications, the cores belong to L p (0, 1) for some p > 1. In this case, we completely describe the eigenvalues of V ϕ and the null space and the range of μI − V ϕ. In addition, a necessary and sufficient condition is given such that μI − V ϕ is a Fredholm operator. In the end, we discuss the regularity of solutions.