Volterra operator

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In mathematics, in the area of functional analysis and operator theory, 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). It is the operator corresponding to the Volterra integral equations.

[edit] Definition

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

[edit] Properties

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

• 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.