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

This paper discusses the two classical Hardy operators ${\mathcal{H}}_1$ on $L^2(0,1)$ and ${\mathcal{H}}_{\mathrm{\infty }}$ on $L^2(0,\mathrm{\infty })$ initially studied by Brown, Halmos and Shields. Particular emphasis is given to the construction of explicit cyclic and $\ast$-cyclic vectors in conjunction with a characterization of their invariant and reducing subspaces. We also provide a complete description of the frame vectors for $I-{\mathcal{H}}_1^{\ast }$.