Frames of orbits of multiplication operators on Hardy spaces

We study frames for Hardy spaces generated by orbits of multiplication operators. We characterize the symbols $\phi \in H^{\mathrm{\infty }}({\mathbb{T}}^N)$ for which the multiplication operator $M_{\phi }$ admits a frame of orbits on $H^2({\mathbb{T}}^N)$. We also show that, in this setting, the existence of a frame is equivalent to the existence of a Parseval frame. Moreover, for $N=1$ we prove that finitely many orbits suffice if and only if $\phi$ is a finite Blaschke product. For $N>1$, no finite collection of orbits can generate a frame, regardless of the symbol. We study the analogous problem for the adjoint operator $M_{\phi }^{\ast }$. Our results extend to the infinite-dimensional torus ${\mathbb{T}}^{\mathrm{\infty }}$ and, via Bohr’s transform, to the Hardy space of Dirichlet series ${\mathcal{H}}_2$.