Volterra operator explained

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^[1]

V(f)(t) = \int ^{t} _{0} f(s) ds.

Properties

• V is a bounded linear operator between Hilbert spaces, with Hermitian adjoint $$V^*(f)(t)\; =\; \backslash int\_^\; f(s)\backslash ,\; ds.$$
• V is a Hilbert–Schmidt operator, hence in particular is compact.^[2]
• V has no eigenvalues and therefore, by the spectral theory of compact operators, its spectrum σ(V) = .^{[2] [3]}
• V is a quasinilpotent operator (that is, the spectral radius, ρ(V), is zero), but it is not nilpotent operator.
• The operator norm of V is exactly ||V|| = ²⁄_π.^[2]

See also

• Product integral

Further reading

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

Notes and References

1. Book: Rynne. Bryan P.. Youngson. Martin A.. Linear Functional Analysis. Springer. 2008. Integral and Differential Equations 8.2. Volterra Integral Equations. 245.
2. Web site: Spectrum of Indefinite Integral Operators . . May 30, 2012 .
3. Web site: Volterra Operator is compact but has no eigenvalue . .