Frame Properties of Operator Orbits

TL;DR

This paper investigates which frames in a Hilbert space can be generated by iterating a linear operator on a single vector, revealing limitations for overcomplete Gabor and wavelet frames and expanding understanding of operator-generated frames.

Contribution

It characterizes frames that can be expressed as operator orbits, especially identifying that overcomplete Gabor and wavelet frames cannot be generated by bounded operators, and explores the broader class of such frames.

Findings

Overcomplete Gabor and wavelet frames cannot be generated by bounded operators.

The class of frames representable as operator orbits with bounded operators is larger than previously thought.

The problem for bi-infinite sequences remains open.

Abstract

We consider sequences in a Hilbert space $\mathcal H$ of the form $(T^nf_0)_{n\in I},$ with a linear operator $T$, the index set being either $I = \mathbb N$ or $I = \mathbb Z$, a vector $f_0\in \mathcal H$, and answer the following two related questions: (a) {\it Which frames for $\mathcal H$ are of this form with an at least closable operator $T$?} and (b) {\it For which bounded operators $T$ and vectors $f_0$ is $(T^nf_0)_{n\in I}$ a frame for $\mathcal H$?} As a consequence of our results, it turns out that an overcomplete Gabor or wavelet frame can never be written in the form $(T^nf_0)_{n\in\mathbb N}$ with a bounded operator $T$. The corresponding problem for $I = \mathbb Z$ remains open. Despite the negative result for Gabor and wavelet frames, the results demonstrate that the class of frames that can be represented in the form $(T^nf_0)_{n\in\mathbb N}$ with a bounded operator $T$ is significantly larger than what could be expected from the examples known so far.