- Volterra operator
In

mathematics , in the area offunctional analysis andoperator theory , the**Volterra operator**represents the operation ofindefinite integration , viewed as abounded linear operator on the space "L"^{2}(0,1) of complex-valuedsquare integrable function s on the interval (0,1). It is the operator corresponding to theVolterra integral equation s.**Definition**The

**Volterra operator**"V" may be defined at a function $x(s)\; in\; L^2\; left(0,\; 1\; ight)$ and a value $t\; in\; left(0,\; 1\; ight)$:$V(x)(t)\; =\; int\_0^t\{x(s),\; ds\}.$

**Properties***"V" is a bounded linear operator between Hilbert spaces, with

Hermitian adjoint ::$V^*(x)(t)\; =\; int\_t^1\{x(s),\; ds\}.$

*"V" is aHilbert-Schmidt operator , hence in particular is compact.

*"V" has noeigenvalue s and therefore, by thespectral theory of compact operators , its spectrum σ("V") = {0}.

*"V" is aquasinilpotent operator (that is, thespectral radius , "ρ"("V"), is zero), but it is notnilpotent .

*Theoperator norm of "V" is exactly ||"V"|| =^{2}⁄_{π}.

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