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 (V _φ)(t):=f^t₀ t^-1φ(st^-1)u(s)ds with the core φ Element 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.