Volterra operator

Example

The Volterra operator is the operator $V$ on the Lp space $L^2(0, 1)$ given by $$(V f)(t) = \int_0^t f(s) \, ds, \quad t \in [0, 1].$$ It is the integral operator of the Volterra equation of the first kind with kernel $K(t, s) = 1$ for $s \le t$ and $K(t, s) = 0$ for $s > t$.

Check that it is a compact operator: the kernel is square-integrable on $[0, 1]^2$, with $\int_0^1 \int_0^1 \abs{K}^2 = 1/2$. So $V$ is a Hilbert–Schmidt integral operator, and Hilbert–Schmidt operators are compact.

Check that $V$ has no eigenvalue. If $V f = 0$, then $\int_0^t f = 0$ for all $t$, so $f = 0$ almost everywhere. If $V f = \lambda f$ with $\lambda \ne 0$, then $f = \lambda^{-1} V f$ is continuous, hence $V f$ is continuously differentiable, and $f' = f / \lambda$ with $f(0) = 0$; so $f = 0$.

Check that the spectrum is $\set{0}$. The power $V^n$ has kernel $(t - s)^{n-1}/(n-1)!$ for $s \le t$, so $\norm{V^n} \le 1/(n-1)!$. Hence the spectral radius $\lim_n \norm{V^n}^{1/n}$ is $0$, while $0$ is in the spectrum because a compact operator on an infinite-dimensional space is not invertible.