Volterra operator

In mathematics, in the area of functional analysis and operator theory, the Volterra operator, named after Vito Volterra, is a bounded linear operator on the space L²[0,1] of complex-valued square-integrable functions on the interval [0,1]. On the subspace C[0,1] of continuous functions it represents indefinite integration. It is the operator corresponding to the Volterra integral equations.

Definition

The Volterra operator, V, may be defined for a function f ∈ L²[0,1] and a value t ∈ [0,1], as

V(f)(t)=\int _{0}^{t}f(s)\,ds.

V is a bounded linear operator between Hilbert spaces, with kernel form $Vf(x)=\int _{0}^{1}1_{y\leq x}f(y)dy$ proven by exchanging the integral sign.
V is a Hilbert–Schmidt operator with norm $\|V\|_{HS}^{2}=1/2$ , hence in particular is compact.
Its Hermitian adjoint has kernel form $V^{*}(f)(x)=\int _{x}^{1}f(y)dy=\int _{0}^{1}1_{y\geq x}f(y)dy$
The positive-definite integral operator $K:=V^{*}V$ has kernel form $Kf(x)=\int _{0}^{1}\min(1-x,1-y)f(y)dy$ proven by exchanging the integral sign. Similarly, $VV^{*}$ has kernel $\min(x,y)$ . They are unitarily equivalent via $Uf(x)=f(1-x)$ , so both have the same spectrum.
The eigenfunctions of $VV^{*}$ satisfy ${\begin{cases}f(1)&=0\\f'(0)&=0\\f''(x)&=-\lambda ^{-1}f\end{cases}}$ with solution $f(x)=\sin((k+1/2)\pi x),\lambda =\left({\frac {1}{(k+1/2)\pi }}\right)^{2}$ with $k=0,1,2,\dots$ .
The singular values of V are $((k+1/2)\pi )^{-1}$ with $k=0,1,2,\dots$ .
The operator norm of V is $2/\pi$ .
V is not trace class.
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 operator.