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 of complex-valued square-integrable functions on the interval . On the subspace C of continuous functions it represents indefinite integration. It is the operator corresponding to the Volterra [integral equation]s.

Definition

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

Properties

V is a bounded linear operator between Hilbert spaces, with kernel form proven by exchanging the integral sign.
V is a Hilbert–Schmidt operator with norm, hence in particular is compact.
Its Hermitian adjoint has kernel form
The positive-definite integral operator has kernel formproven by exchanging the integral sign. Similarly, has kernel. They are unitarily equivalent via, so both have the same spectrum.
The eigenfunctions of satisfy with solution with.
The singular values of V are with.
The operator norm of V is.
V is not trace class.
V has no eigenvalues and therefore, by the spectral theory of compact operators, its spectrum σ =.
V is a quasinilpotent operator, but it is not nilpotent operator.