Bessel orbits of normal operators

TL;DR

This paper characterizes when the sequence generated by powers of a bounded normal operator forms a Bessel sequence, with explicit conditions for special cases and applications to heat equation operators, advancing the understanding in dynamical sampling.

Contribution

It provides a spectral measure-based characterization of Bessel sequences generated by normal operators, including explicit results for unitary, selfadjoint, and heat equation-related operators.

Findings

Characterization in terms of spectral measure for normal operators

Explicit conditions for unitary and selfadjoint cases

Application to operators from the heat equation

Abstract

Given a bounded normal operator $A$ in a Hilbert space and a fixed vector $x$, we elaborate on the problem of finding necessary and sufficient conditions under which $(A^kx)_{k\in\mathbb N}$ constitutes a Bessel sequence. We provide a characterization in terms of the measure $\|E(\cdot)x\|^2$, where $E$ is the spectral measure of the operator $A$. In the separately treated special cases where $A$ is unitary or selfadjoint we obtain more explicit characterizations. Finally, we apply our results to a sequence $(A^kx)_{k\in\mathbb N}$, where $A$ arises from the heat equation. The problem is motivated by and related to the new field of Dynamical Sampling which was recently initiated by Aldroubi et al.