# 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.

## Definition

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

$V(f)(t) = \int_0^t{f(s)\, ds}.$

## Properties

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