Multi-orbital frames through model spaces

TL;DR

This paper characterizes when sequences generated by normal operators on  and specific vectors form frames, using complex analysis tools like the pseudo-hyperbolic metric and Blaschke products.

Contribution

It provides a novel characterization of frame sequences generated by normal operators using model spaces and complex function theory.

Findings

Characterization of frame sequences via model spaces and Blaschke products

Use of pseudo-hyperbolic metric in the analysis

Connection between operator-generated sequences and invariant subspaces

Abstract

We characterize the normal operators $A$ on $\ell^2$ and the elements $a^i \in \ell^2$, with $1\le i\le m$, such that the sequence $$\{ A^n a^1 , \ldots , A^n a^m \}_{n\ge 0}$$ is a frame. The characterization makes strong use of the pseudo-hyperbolic metric of $\mathbb{D}$ and is given in terms of the backward shift invariant subspaces of $H^2(\mathbb{D})$ associated to finite products of interpolating Blaschke products.