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

TL;DR

This work provides explicit, function-space descriptions of invariant and reducing subspaces, star-cyclic and cyclic vectors, and frame vectors for the classical Hardy integral operators $\mathcal{H}_1$ on $L^2(0,1)$ and $\mathcal{H}_{\infty}$ on $L^2(0,\infty)$. Central to the approach is a Mellin-transform framework that realizes these operators as multiplication operators $M_{\varphi}$ and Laguerre-shift models, yielding concrete descriptions of subspaces via Beurling–Wiener theory and explicit bases from Laguerre polynomials. The paper establishes a complete description of reducing subspaces for $\mathcal{H}_{\infty}$, characterizes $\ast$-cyclic vectors, connects invariant-subspace structures to semigroups of dilations, and clarifies the commutants and frame-vector phenomena (notably that $I-\mathcal{H}_1^{*}$ has abundant frame vectors while $I-\mathcal{H}_1$ and $I-\mathcal{H}_{\infty}$ do not). These results extend classical shift-model perspectives to explicit, transform-based descriptions in $L^2$-spaces and enrich the operator-theoretic understanding of Hardy-type integral operators.

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 $I - \mathcal{H}_{1}^{*}$.

Theorems & Definitions (33)

• Proposition 2.6
• proof
• Lemma 3.1
• Proposition 3.8
• proof
• Proposition 3.11
• Proposition 3.13
• Proposition 3.14
• Corollary 3.15
• Proposition 3.18