Abstract
We call a Volterra integral operator of the form , 0 ≤ t ≤ T, ϕ ∈ L 1(0, 1), to be cordial and the function ϕ its core. The class of cordial Volterra operators contains Diogo's, Lighthill's and other noncompact weakly singular Volterra integral operators occuring in the literature. Using well known results from the theory of commutative Banach algebras, we derive a formula for the spectrum σ m (V ϕ) of V ϕ considered as an operator in the space C m [0, T], m = 0, 1,…. A cordial operator V ϕ has the property that the functions t r , 0 ≤ r < ∞, are eigenfunctions of V ϕ corresponding to the eigenvalues ϕ(s)s r ds. This enables a simple solving of the cordial equation μu = V ϕ u + f approximating f by a polynomial or, when possible, expanding f into a power or generalized power series.
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