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

TL;DR

This paper characterizes invariant subspaces of the Cesàro operator on Hardy spaces using shift semigroups, linking them to composition operators, and extends results on the operator's square root.

Contribution

It provides a novel characterization of Cesàro operator invariant subspaces via shift semigroups and composition operators, and extends the understanding of the operator's square root.

Findings

Invariant subspaces correspond to those invariant under specific composition semigroups.

Complete characterization of finite codimensional invariant subspaces.

Extension of results on the square root of the Cesàro operator.

Abstract

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.