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^[1]

V (f) (t) = \int_{0}^{t} f (s) d s .

V is a bounded linear operator between Hilbert spaces, with kernel form $V f (x) = \int_{0}^{1} 1_{y \leq x} f (y) d y$ proven by exchanging the integral sign.
V is a Hilbert–Schmidt operator with norm $‖ V ‖_{H S}^{2} = 1 / 2$ , hence in particular is compact.
Its Hermitian adjoint has kernel form $V^{*} (f) (x) = \int_{x}^{1} f (y) d y = \int_{0}^{1} 1_{y \geq x} f (y) d y$
The positive-definite integral operator $K : = V^{*} V$ has kernel form $K f (x) = \int_{0}^{1} \min (1 - x, 1 - y) f (y) d y$ proven by exchanging the integral sign. Similarly, $V V^{*}$ has kernel $\min (x, y)$ . They are unitarily equivalent via $U f (x) = f (1 - x)$ , so both have the same spectrum.
The eigenfunctions of $V V^{*}$ satisfy ${\begin{cases} f (0) & = 0 \\ f^{'} (1) & = 0 \\ f^{″} (x) & = - λ^{- 1} f \end{cases}$ with solution $f (x) = \sin ((k + 1 / 2) π x), λ = {(\frac{1}{(k + 1 / 2) π})}^{2}$ with $k = 0, 1, 2, \dots$ .
The singular values of V are $((k + 1 / 2) π)^{- 1}$ with $k = 0, 1, 2, \dots$ .
The operator norm of V is $2 / π$ .
V is not trace class.
V has no eigenvalues and therefore, by the spectral theory of compact operators, its spectrum σ(V) = {0}.^[2]^[3]
V is a quasinilpotent operator (that is, the spectral radius, ρ(V), is zero), but it is not nilpotent operator.