Characterization of the Volterra operator and the Riemann-Liouville semigroup

The Volterra operator V : f(x) → R ^x ₀ f(t) dt on C[0, 1] or L^p(0, 1) (1 ≤ p < ∞) is characterized as the unique bounded linear operator on the space satisfying the algebraic condition (∗) [S, V ] = V ², V e = Se, where S is the multiplication operator f(x) → xf(x), and e is the function with constant value 1. Similarly, if S_α is the multiplication operator f → α f on C[0, 1], where α is a given injective real-valued C[0, 1]-function of bounded variation vanishing at 0, then the Stieltjes-Volterra operator f(x) → R ^x ₀ f(t) dα(t) on C[0, 1] is characterized as the unique bounded linear operator on the space satisfying the above condition with S = S _α. For 1 < p < ∞, the Riemann-Liouville semigroup is characterized as the unique regular semigroup V (·) on ℂ⁺ acting in L^p(0, 1), whose boundary group's type is less than π, and for which V := V (1) satisfies Relation (∗).