Invariant subspaces of the Cesaro operator

TL;DR

This paper characterizes invariant subspaces of the Cesàro operator on Hardy space, identifying finite co-dimensional and model space invariants, and explores their structure and connections to composition operators.

Contribution

It provides a new characterization of finite co-dimensional invariant subspaces and identifies which model spaces are invariant under the Cesàro operator.

Findings

Finite co-dimensional invariant subspaces characterized.

Model spaces invariant under Cesàro operator identified.

Invariant subspaces within model spaces are cyclic.

Abstract

This paper explores various classes of invariant subspaces of the classical Ces\`{a}ro operator $C$ on the Hardy space $H^2$. We provide a new characterization of the finite co-dimensional $C$-invariant subspaces, based on earlier work of the first two authors, and determine exactly which model spaces are $C$-invariant subspaces. We also describe the $C$-invariant subspaces contained in model spaces and establish that they are all cyclic. Along the way, we re-examine an associated Hilbert space of analytic functions on the unit disk developed by Kriete and Trutt. We also make a connection between the adjoint of the Ces\`{a}ro operator and certain composition operators on $H^2$ which have universal translates in the sense of Rota.