# Volterra operator

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A completely-continuous linear operator $V$ (cf. Completely-continuous operator), acting on a Banach space, whose spectrum consists of the point zero only. An example of a linear Volterra integral operator on the space of functions which are square-summable on $[a,b]$ is

$$V\phi(x)=\int\limits_a^x K(x,s)\phi(s)ds.$$

A non-linear Volterra integral operator is an operator of the form

$$V\phi(x)=\int\limits_a^x K(x,s,\phi(s))ds.$$

Named after V. Volterra, who studied the Volterra integral equations corresponding to such operators (cf. Volterra equation).

#### Comments

The spectral theory of Volterra operators on Hilbert space (invariant subspaces, canonical model, unitary invariants) is an important topic in the theory of non-self-adjoint operators. Since the spectrum consists of one point only, the classical spectral methods from the theory of self-adjoint operators are not applicable to Volterra operators, and new tools are used to study such operators, among others the theory of characteristic operator functions. See [a1], [a2] for further information. Volterra operators are also used to provide mathematical models for problems of population dynamics [a3]. See [a4] for the general theory of Volterra integral and functional equations.

How to Cite This Entry:
Volterra operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Volterra_operator&oldid=31954
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article