Hardy operators: In the footsteps of Brown, Halmos, and Shields

TL;DR

This paper explores classical Hardy operators, focusing on their invariant subspaces, cyclic vectors, and frame vectors, extending foundational work by Brown, Halmos, and Shields.

Contribution

It provides explicit constructions of cyclic and $*$-cyclic vectors and characterizes invariant, reducing, and frame subspaces for Hardy operators.

Findings

Explicit cyclic and $*$-cyclic vectors constructed.

Complete description of invariant and reducing subspaces.

Characterization of frame vectors for $I - \\mathcal{H}_{1}^{*}$.

Abstract

This paper discusses the two classical Hardy operators $\mathcal{H}_{1}$ on $L^2(0, 1)$ and $\mathcal{H}_{\infty}$ on $L^2(0, \infty)$ initially studied by Brown, Halmos and Shields. Particular emphasis is given to the construction of explicit cyclic and $*$-cyclic vectors in conjunction with a characterization of their invariant and reducing subspaces. We also provide a complete description of the frame vectors for $I - \mathcal{H}_{1}^{*}$.