View source for Volterra operator

A completely-continuous linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096860/v0968601.png" /> (cf. [[Completely-continuous operator|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096860/v0968602.png" /> is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096860/v0968603.png" /></td> </tr></table>

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

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Named after V. Volterra, who studied the Volterra integral equations corresponding to such operators (cf. [[Volterra equation|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 [[#References|[a1]]], [[#References|[a2]]] for further information. Volterra operators are also used to provide mathematical models for problems of population dynamics [[#References|[a3]]]. See [[#References|[a4]]] for the general theory of Volterra integral and functional equations.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.S. Brodskii,   "Triangular and Jordan representations of linear operators" , Amer. Math. Soc.  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.A.J. Metz (ed.)  O. Diekmann (ed.) , ''The dynamics of physiologically structured populations'' , ''Lect. notes in biomath.'' , '''68''' , Springer  (1986)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Grippenberg,   S.-O. Londen,   O. Staffans,   "Volterra integral and functional equations" , Cambridge Univ. Press  (1990)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.L. Moiseiwitsch,   "Integral equations" , Longman  (1977)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.J. Jerri,   "Introduction to integral equations with applications" , M. Dekker  (1985)  pp. Sect. 2.3</TD></TR></table>