# Abstract Volterra operator

An operator $V$ acting between function spaces $E([t_0,t_1),S)$ and $F([t_0,t_1),T)$, such that $x(s)=y(s)$ on $t_0\leq s\leq t\leq t_1$ implies $(Vx)(s)=(Vy)(s)$ on $t_0\leq s\leq t<t_1$ for any $t<t_1$. If one deals with spaces of measurable functions, then the equalities above must be satisfied almost everywhere. In the engineering literature such operators are called causal operators or non-anticipative operators. The classical Volterra integral operator (cf. Volterra operator) is the example encountered most often. These operators occur in the description of phenomena or processes in which the future evolution is influenced by the past. The idea of an abstract Volterra operator appears in V. Volterra's work quite clearly, even though a formal definition and results are missing. The first paper on abstract Volterra operators was by L. Tonelli [a10], in which the idea was used to prove existence theorems for equations of the form $x(t)=(Vx)(t)$. In [a14] A.N. Tykhonov also considered this concept and stressed its importance in applications. The first book dealing with such operators was [a6]. L. Neustadt has shown the significance of these operators in control theory [a6].

A remarkable result concerning the connection between classical and abstract Volterra operators was given by I.W. Sandberg [a12]. Under suitable conditions, abstract Volterra operators can be approximated to any degree of accuracy by means of Volterra series (i.e., by means of classical Volterra operators in integral form). The abstract Volterra operators on a given function space can be organized as an algebra, since a sum or product (superposition) of such operators is again an abstract Volterra operator. The inverse, if it exists, is not necessarily an abstract Volterra operator (a simple example is given by $(Vx)(t)=x(t/2)$ when the interval of definition for the functions is the positive semi-axis).

Various properties of abstract Volterra operators and their use in many branches of applied science can be found in [a1], [a2], [a3], [a4], [a6], [a7], [a9], [a13]. The survey paper [a5] contains recent results on abstract Volterra operators and associated equations. However, a solid theory of this kind of operator does not yet exist.

There are at least two more meanings of the term "abstract Volterra operator" . First, a linear operator on an abstract space whose spectrum reduces to the unique point zero is sometimes called an abstract Volterra operator, see [a15]. A second use of the term is when the function spaces involved consist of functions taking values in an abstract (Banach or Hilbert) space. Such operators are widely encountered in continuum mechanics. In [a8], problems in viscoelasticity leading to this type of Volterra operators are studied.

#### References

[a1] | N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, "Introduction to the theory of functional differential equations" , Nauka (1991) (In Russian) |

[a2] | V. Barbu, "Nonlinear semigroups and differential equations in Banach spaces" , Noordhoff (1976) |

[a3] | A.L. Bugheim, "Introduction to the theory of inverse problems" , Nauka (1988) (In Russian) |

[a4] | C. Corduneanu, "Integral equations and applications" , Cambridge Univ. Press (1991) |

[a5] | C. Corduneanu, "Equations with abstract Volterra operators and their control" , Ordinary Differential Equations and their Applications , Firenze–Bologna (1995) |

[a6] | L. Neustadt, "Optimization (a theory of necessary conditions)" , Princeton Univ. Press (1976) |

[a7] | J. Pruss, "Evolutionary integral equations" , Birkhäuser (1993) |

[a8] | M. Renardy, W.J. Hrusa, J.A. Nohel, "Mathematical problems in viscoelasticity" , Longman (1987) |

[a9] | G. Gripenberg, S.O. Londen, O. Staffans, "Volterra integral and functional equations" , Cambridge Univ. Press (1990) |

[a10] | L. Tonelli, "Sulle equazioni funzionali di Volterra" Bull. Calcutta Math. Soc. , 20 (1929) |

[a11] | V. Volterra, "Opere Matematiche" , 1–3 , Accad. Naz. Lincei (1954–1955) |

[a12] | I.W. Sandberg, "Expansions for nonlinear systems, and Volterra expansions for time-varying nonlinear systems" Bell System Techn. J. , 61 (1982) pp. 159–225 |

[a13] | M. Schetzen, "The Volterra and Wiener theories of nonlinear systems" , Wiley (1980) |

[a14] | A.N. Tychonoff, "Sur les équations fonctionnelles de Volterra et leurs applications à certains problèmes de la physique mathématique" Bull. Univ. Moscou Ser. Internat. , A1 : 8 (1938) |

[a15] | I.C. Gokhberg, M.G. Krein, "Theory of Volterra operators in Hilbert space and its applications" , Nauka (1967) (In Russian) |

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Abstract Volterra operator.

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