Iterative actions of normal operators

TL;DR

This paper explores the conditions under which iterative systems generated by normal operators and a countable set of vectors form complete, Bessel, basis, or frame systems in a Hilbert space, linking spectral theory and dynamical sampling.

Contribution

It characterizes the relations between normal operators, vector sets, and iteration lengths that produce various structured systems in Hilbert spaces, advancing understanding in dynamical sampling and frame theory.

Findings

Conditions for completeness and basis formation identified

Connections established between spectral properties and iterative systems

Applications to wavelet and time-frequency analysis explored

Abstract

Let $A$ be a normal operator in a Hilbert space $\mathcal{H}$, and let $\mathcal{G} \subset \mathcal{H}$ be a countable set of vectors. We investigate the relations between $A$, $\mathcal{G}$ , and $L$ that makes the system of iterations $\{A^ng: g\in \mathcal{G},\;0\leq n< L(g)\}$ complete, Bessel, a basis, or a frame for $\mathcal{H}$. The problem is motivated by the dynamical sampling problem and is connected to several topics in functional analysis, including, frame theory and spectral theory. It also has relations to topics in applied harmonic analysis including, wavelet theory and time-frequency analysis.