Volterra operator

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

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

## Properties

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* V is a bounded linear operator between Hilbert spaces, with Hermitian adjoint V ∗ ( f ) ( t ) = ∫ t 1 f ( s ) d s . {\displaystyle V^{*}(f)(t)=\int _{t}^{1}f(s)\,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) = {0}.[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

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* Product integral

## References

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1. ^ Rynne, Bryan P.; Youngson, Martin A. (2008). "Integral and Differential Equations 8.2. Volterra Integral Equations". Linear Functional Analysis. Springer. p. 245.
2. ^ a b c "Spectrum of Indefinite Integral Operators". Stack Exchange. May 30, 2012.
3. ^ "Volterra Operator is compact but has no eigenvalue". Stack Exchange.

## Further reading

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* Gohberg, Israel; Krein, M. G. (1970). Theory and Applications of Volterra Operators in Hilbert Space. Providence: American Mathematical Society. ISBN 0-8218-3627-7.