Properties of convolution operators on Lp(0,1)

Convolution operators on Lp(0,1) have many similarities with the classical Volterra operator V, but it is not known in general for which convolution kernels the resulting operator behaves like V.

It is shown that many convolution operators are cyclic, and the cyclic property is related to the invariant subspace lattice of the operator, and to the behavior of the kernel as an element of the Volterra algebra.

The convolution operators induced by kernels satisfying a smoothness condition near the origin are shown to have asymptotic behavior that matches that of powers of V, and a new class of convolution operators that are not nilpotent, but have kernels that are not polynomial generators for L1(0,1), are produced.

For kernels that are polynomial generators for L1(0,1), the corresponding convolution operators are shown to have the property that their commutant and the strongly-closed subalgebra of B(Lp(0,1)) they generate are equal.