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

Properties

V is a bounded linear operator between Hilbert spaces, with kernel formV f ( x ) = ∫ 0 1 1 y ≤ x f ( y ) d y {\displaystyle 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 ‖ H S 2 = 1 / 2 {\displaystyle \|V\|_{HS}^{2}=1/2}, hence in particular is compact.
Its Hermitian adjoint has kernel formV ∗ ( f ) ( x ) = ∫ x 1 f ( y ) d y = ∫ 0 1 1 y ≥ x f ( y ) d y {\displaystyle 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 {\displaystyle K:=V^{*}V} has kernel formK f ( x ) = ∫ 0 1 min ( 1 − x , 1 − y ) f ( y ) d y {\displaystyle Kf(x)=\int _{0}^{1}\min(1-x,1-y)f(y)dy}proven by exchanging the integral sign. Similarly, V V ∗ {\displaystyle VV^{*}} has kernel min ( x , y ) {\displaystyle \min(x,y)}. They are unitarily equivalent via U f ( x ) = f ( 1 − x ) {\displaystyle Uf(x)=f(1-x)}, so both have the same spectrum.
The eigenfunctions of V V ∗ {\displaystyle VV^{*}} satisfy { f ( 1 ) = 0 f ′ ( 0 ) = 0 f ″ ( x ) = − λ − 1 f {\displaystyle {\begin{cases}f(1)&=0\\f'(0)&=0\\f''(x)&=-\lambda ^{-1}f\end{cases}}} with solution f ( x ) = sin ⁡ ( ( k + 1 / 2 ) π x ) , λ = ( 1 ( k + 1 / 2 ) π ) 2 {\displaystyle f(x)=\sin((k+1/2)\pi x),\lambda =\left({\frac {1}{(k+1/2)\pi }}\right)^{2}}with k = 0 , 1 , 2 , … {\displaystyle k=0,1,2,\dots }.
The singular values of V are ( ( k + 1 / 2 ) π ) − 1 {\displaystyle ((k+1/2)\pi )^{-1}} with k = 0 , 1 , 2 , … {\displaystyle k=0,1,2,\dots }.
The operator norm of V is 2 / π {\displaystyle 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.

Volterra operator

Definition

Properties

See also

Further reading