# Volterra operator

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.

#### References

[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.

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