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).


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.


[a1] I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Theory and applications of Volterra operators in Hilbert space" , Amer. Math. Soc. (1970) (Translated from Russian)
[a2] M.S. Brodskii, "Triangular and Jordan representations of linear operators" , Amer. Math. Soc. (1971) (Translated from Russian)
[a3] J.A.J. Metz (ed.) O. Diekmann (ed.) , The dynamics of physiologically structured populations , Lect. notes in biomath. , 68 , Springer (1986)
[a4] G. Grippenberg, S.-O. Londen, O. Staffans, "Volterra integral and functional equations" , Cambridge Univ. Press (1990)
[a5] B.L. Moiseiwitsch, "Integral equations" , Longman (1977)
[a6] A.J. Jerri, "Introduction to integral equations with applications" , M. Dekker (1985) pp. Sect. 2.3
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Volterra operator. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article