Frames via Unilateral Iterations of a Bounded Operator

TL;DR

This paper characterizes when a frame in an infinite-dimensional Hilbert space can be represented as iterates of a bounded operator applied to a vector, advancing understanding in dynamical sampling and operator theory.

Contribution

It provides necessary and sufficient conditions for frames to be expressed as unilateral iterations of a bounded operator, and characterizes vectors generating such frames.

Findings

Established criteria for frames to be of the form {T^n φ}

Characterized vectors φ that generate frames via operator iterations

Included auxiliary results on Riesz frame operator representations

Abstract

Motivated by recent work in Dynamical Sampling, we prove a necessary and sufficient condition for a frame in a separable and infinite-dimensional Hilbert space to admit the form $\{T^{n} \varphi \}_{n \geq 0}$ with $T \in B(H)$. Also, a characterization of all the vectors $\varphi$ for which $\{T^{n} \varphi \}_{n \geq 0}$ is a frame for some $T \in B(H)$ is provided. Some auxiliary results on operator representations of Riesz frames are given as well.