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

[math]\displaystyle{ V(f)(t) = \int_{0}^{t} f(s)\, ds. }[/math]

Properties

• V is a bounded linear operator between Hilbert spaces, with Hermitian adjoint [math]\displaystyle{ V^*(f)(t) = \int_{t}^{1} f(s)\, ds. }[/math]
• 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.

Further reading

• Gohberg, Israel; Krein, M. G. (1970). Theory and Applications of Volterra Operators in Hilbert Space. Providence: American Mathematical Society. ISBN 0-8218-3627-7.