Insights on the Ces\`aro operator: shift semigroups and invariant subspaces

A closed subspace is invariant under the Ces\`aro operator $\mathcal{C}$ on the classical Hardy space $H^2(\mathbb D)$ if and only if its orthogonal complement is invariant under the $C_0$-semigroup of composition operators induced by the affine maps $\varphi_t(z)= e^{-t}z + 1 - e^{-t}$ for $t\geq 0$ and $z\in \mathbb D$. The corresponding result also holds in the Hardy spaces $H^p(\mathbb D)$ for $1<p<\infty$. Moreover, in the Hilbert space setting, by linking the invariant subspaces of $\mathcal{C}$ 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 $\mathcal{C}$. 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 $\mathcal{C}$, and discuss its invariant subspaces.