Volterra operator

In mathematics, in the area of functional analysis and operator theory, the Volterra operator, named after Vito Volterra, is a bounded linear operator on the space L²(0,1) of complex-valued square-integrable functions on the interval (0,1). On the subspace C(0,1) of continuous functions it represents indefinite integration. It is the operator corresponding to the Volterra integral equations.

Definition

The Volterra operator, V, may be defined for a function f ∈ L²(0,1) and a value t ∈ (0,1), as

$V(f)(t)=\int _{0}^{t}f(s)\,ds.$

Properties

• V is a bounded linear operator between Hilbert spaces, with Hermitian adjoint

$V^{*}(f)(t)=\int _{t}^{1}f(s)\,ds.$

• V is a Hilbert–Schmidt operator, hence in particular is compact.
• V has no eigenvalues and therefore, by the spectral theory of compact operators, its spectrum σ(V) = {0}.
• V is a quasinilpotent operator (that is, the spectral radius, ρ(V), is zero), but it is not nilpotent.
• The operator norm of V is exactly ||V|| = ²⁄_π.