Dynamical Frames and Hyperinvariant Subspaces
Victor Bailey, Deguang Han, Keri Kornelson, David Larson, Rui Liu
TL;DR
The paper characterizes central frame representations for arbitrary semigroups using co-hyperinvariant subspaces of the left regular representation. It shows that when the wot-closed semigroup algebra $\mathcal{A}_{\mathcal{S}}$ is maximal abelian, every frame representation is central, and it applies this to commuting $k$-tuples $\{A_1^{n_1}\cdots A_k^{n_k}\xi\}$ on a separable Hilbert space. In particular, for $\mathcal{S} = \mathbb{Z}_{+}^k$, the multiplier algebra $\mathcal{M}(\mathbb{D}^k)$ is maximal abelian, yielding equivalence of all such dynamical frames; this extends to hybrid cases $\mathcal{G} \times \mathbb{Z}_{+}^k$ via tensor products. The results unify and extend the group-case Han-Larson theory to a broad class of semigroups and provide a concrete criterion for centrality via invariant-subspace structure.
Abstract
The theory of dynamical frames evolved from practical problems in dynamical sampling where the initial state of a vector needs to be recovered from the space-time samples of evolutions of the vector. This leads to the investigation of structured frames obtained from the orbits of evolution operators. One of the basic problems in dynamical frame theory is to determine the semigroup representations, which we will call central frame representations, whose frame generators are unique (up to equivalence). Recently, Christensen, Hasannasab, and Philipp proved that all frame representations of the semigroup $\Bbb{Z}_{+}$ have this property. Their proof of this result relies on the characterization of the structure of shift-invariant subspaces in $H^2(\mathbb{D})$ due to Beurling. In this paper we settle the general uniqueness problem by presenting a characterization of central frame representations for any semigroup in terms of the co-hyperinvariant subspaces of the left regular representation of the semigroup. This result is not only consistent with the known result of Han-Larson in 2000 for group representation frames, but also proves that all the frame generators of a semigroup generated by any $k$-tuple $(A_1, ... A_k)$ of commuting bounded linear operators on a separable Hilbert space $H$ are equivalent, a case where the structure of shift-invariant subspaces, or submodules, of the Hardy Space on polydisks $H^{2}(\Bbb{D}^k)$ is still not completely characterized.