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Volterra operator

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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.

Contents

Definition

The Volterra operator, V, may be defined for a function f ∈ L2(0,1) and a value t ∈ (0,1), as

V ( f ) ( t ) = ∫ 0 t f ( s ) d s .

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.

    • References

    Volterra operator Wikipedia
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