Convolution operators on Lebesgue spaces of the half-line

Abstract:

In this paper we determine the lattice of closed invariant subspaces for certain convolution operators on Lebesgue spaces ${L^p}(d\sigma )$ where $\sigma$ is a suitable weighted measure on the half-line. We exploit the rather close relationship between convolution operators and the collection of right translation operators ${\{ {T_\lambda }\} _{\lambda \geqq 0}}$ on ${L^p}(d\sigma )$. We show that a convolution operator K and the collection ${\{ {T_\lambda }\} _{\lambda \geqq 0}}$ have the same lattice of closed invariant subspaces provided the kernel k of K is a cyclic vector. The converse also holds if we assume in addition that the closed span of ${\{ {T_\lambda }k\} _{\lambda \geqq 0}}$ is all of ${L^p}(d\sigma )$. We show that the lattice of closed right translation invariant subspaces of ${L^p}(d\sigma )$ is totally ordered by set inclusion whenever $\sigma$ has compact support. Thus in this case a convolution operator K is unicellular if and only if its kernel is a cyclic vector. Finally, we show for suitable weighted measures $\sigma$ on the half-line that the convolution operators on ${L^p}(d\sigma )$ are Volterra.