On invariant subspaces of Volterra-type operators

The lattice of all the closed, invariant subspaces of the Volterra integration operator onL2[0, 1] is equal to {B(a):a∈[0, 1]}, whereB(a)={f∈L2[0, 1]:f=0 a.e. on [0,a]}. In order to extend this result to Banach function spaces we study the Volterra-type operatorV that was introduced in [7] for the case ofLp-spaces. Our main result characterizesL′-closed subspaces of a Banach function spaceL that are invariant underV, whereL′ denotes the associate space ofL. In particular, if the norm ofL is order continuous and ifV is injective, then all the closed, invariant subspaces ofV are determined.