Abstract
Given a Lebesgue measurable self-map \({\varphi}\) of the interval [0, 1], the Volterra- composition operator is defined as \({(V_\varphi)(x)=\int\limits_0^{\varphi(x)}f(t)\,dt, \quad f \in L^p[0,1],\,\, 1\leq p \leq \infty.}\) We develop the spectral theory of these operators. In particular, for a class of natural symbols \({\varphi}\) , finiteness of the spectrum is characterized and formulae for the trace and the convergence exponent of eigenvalues are provided. The positivity of the spectrum as well as the analyticity of the eigenfunctions are also treated. The theory of entire functions as well as solving some Cauchy Problems will play a fundamental role in this theory. We also supply some examples of symbols \({\varphi}\) to which the theory can be applied and, in particular, eigenvalues and eigenfunctions are computed explicitly.
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