Volterra operator

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In mathematics, in the area of functional analysis and operator theory, the Volterra operator represents the operation of indefinite integration, viewed as a bounded linear operator on the space L²(0,1) of complex-valued square integrable functions on the interval (0,1). It is the operator corresponding to the Volterra integral equations.

Definition

The Volterra operator V may be defined at a function $x(s) \in L^2 \left(0, 1 \right)$ and a value $t \in \left(0, 1 \right)$

$V(x)(t) = \int_0^t{x(s)\, ds}.$

$V^*(x)(t) = \int_t^1{x(s)\, ds}.$

V is a Hilbert-Schmidt operator, hence in particular is compact.
V has no eigenvalues and therefore, by the spectral theory of compact operators, its spectrum σ(V) = {0}.
V is a quasinilpotent operator (that is, the spectral radius, ρ(V), is zero), but it is not nilpotent.
The operator norm of V is exactly ||V|| = ²⁄_π.

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