Volterra operator

In the branch of mathematics known as functional analysis, the Volterra operator represents the operation of indefinite integration, viewed as a bounded linear operator on the space L²(0,1) of complex-valued square integrable functions on the interval (0,1).

Definition

The Volterra operator V may be defined at a function x(s) ∈ L²(0, 1) and a value t ∈ (0, 1) by

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

Properties

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

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

• V is a Hilbert-Schmidt operator, hence in particular is compact.

• V has no eigenvalues and therefore by the Riesz spectral theorem its spectrum σ(V) = {0}.

• V is a quasi-nilpotent operator i.e. the spectral radius ρ(V) = 0, but it is not nilpotent.

• The operator norm of V is exactly ||V|| = ²⁄_π.