Volterra Operator
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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

The Volterra operator, V, may be defined for a function f ? L2[0,1] and a value t ? [0,1], as

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

## Properties

${\displaystyle V^{*}(f)(t)=\int _{t}^{1}f(s)\,ds.}$

## References

1. ^ a b c "Spectrum of Indefinite Integral Operators". Stack Exchange. May 30, 2012.

## Further reading

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

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