Volterra operator
In mathematics, in the area of functional analysis and operator theory, the Volterra operator, named after Vito Volterra, 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). 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, with Hermitian adjoint
- V is a Hilbert–Schmidt operator, hence in particular is compact.[1]
- V has no eigenvalues and therefore, by the spectral theory of compact operators, its spectrum σ(V) = {0}.[1]
- 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.[1]