The lattices of invariant subspaces of a class of operators on the Hardy space

TL;DR

This paper extends previous work to describe the lattice of invariant subspaces for a class of operators on the Hardy space, specifically involving the shift and scaled complex Volterra operators, building on earlier algebraic characterizations.

Contribution

It provides a detailed description of invariant subspaces for the shift plus a scaled complex Volterra operator, extending prior results to a broader class of operators.

Findings

Characterization of invariant subspaces for the scaled operator

Extension of algebraic methods to new operator classes

Connection to previous real-operator studies

Abstract

In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator.