Insights on the Cesaro operator: shift semigroups and invariant subspaces

A closed subspace is invariant under the Cesaro operator C on the classical Hardy space H-2(D) if and only if its orthogonal complement is invariant under the C-0-semigroup of composition operators induced by the affine maps phi(t)(z)=e(-t)z+1-e(-t) for t >= 0 and z is an element of D. The corresponding result also holds in the Hardy spaces H-p(D) for 1 < p < infinity. Moreover, in the Hilbert space setting, by linking the invariant subspaces of C to the lattice of the closed invariant subspaces of the standard right-shift semigroup acting on a particular weighted L-2-space on the line, we exhibit a large class of non-trivial closed invariant subspaces and provide a complete characterization of the finite codimensional ones, establishing, in particular, the limits of such an approach towards describing the lattice of all invariant subspaces of C. Finally, we present a functional calculus argument which allows us to extend a recent result by Mashreghi, Ptak and Ross regarding the square root of C and discuss its invariant subspaces.