Hilbert space models and Blaschke frames

TL;DR

This paper explores the structure and redundancy properties of Blaschke frames in Hilbert spaces, revealing conditions under which these frames are bases, have some redundancy, or are fully redundant, using Hilbert space models and Wold decomposition.

Contribution

It provides a complete characterization of redundancy in Blaschke frames, connecting frame properties with the choice of Blaschke product and isometry, and employs Hilbert space models and Wold decomposition.

Findings

Blaschke frames can be Riesz bases, have some redundancy, or be fully redundant.

Redundancy depends on the specific Blaschke product and isometry used.

The paper offers a complete solution to the redundancy question in this context.

Abstract

For a finite Blaschke product $B$ and for an isometry $V$ on an infinite-dimensional separable complex Hilbert space $\mathcal{H}$ we study a sequence $(b_m)_{m=1}^\infty$ of vectors in $\mathcal{H}$, defined by $b_m = B(V^*)e_m$, where $(e_m)_{m=1}^\infty$ is an orthonormal basis in $\mathcal{H}$. We call $(b_m)_{m=1}^\infty$ a Blaschke frame for $B$ with isometry $V$ on $\mathcal{H}$. We show how instrumental the use of Hilbert space models are in frame theory by completely solving the question of redundancy for a Blaschke frame, that is, what vectors can be removed from the frame $(b_{m})_{m=1}^{\infty}$ such that $(b_{m})_{m\neq k}$ is still a frame? Using the Wold decomposition, we prove that a Blaschke frame can have no redundant vectors (a Riesz basis), have some redundant vectors, or every vector is redundant (a fully insured frame). These unique cases depend on the choice of finite Blaschke product and which isometry one chooses in the construction of a Blaschke frame.