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

Properties

• V is a bounded linear operator between Hilbert spaces proven by exchanging the integral sign.

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

• Its Hermitian adjoint

• . They are unitarily equivalent, so both have the same spectrum.

• .

• The singular values of V.

• The operator norm of V.

• V is not trace class.

• 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 operator.

See also

• Product integral

References

Further reading