Invariant subspaces of the shift plus complex Volterra operator

TL;DR

This paper characterizes the lattice of closed invariant subspaces of the shift plus complex Volterra operator on the Hardy space, extending Sarason's previous results on related operators.

Contribution

It provides a complete description of the invariant subspaces for the shift plus complex Volterra operator on the Hardy space, building on Sarason's work on similar operators.

Findings

Lattice of invariant subspaces explicitly described

Extension of Sarason's results to complex Volterra operator

New structural insights into operator invariant subspaces

Abstract

The lattice of closed invariant subspaces of the Volterra operator acting on $L^2(0,1)$ was completely described by Sarason. On the other hand, he explicitly found the lattice of closed invariant subspaces of the shift plus Volterra operator on $L^2(0,1)$. Inspired by Sarason's results, we find the lattice of closed invariant subspaces of the shift plus complex Volterra operator acting on the Hardy space