Bessel orbits of normal operators

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 $(A^{k}x{)}_{k\in \mathbb{N}}$ constitutes a Bessel sequence. We provide a characterization in terms of the measure $\mathrm{\parallel }E(\cdot )x{\mathrm{\parallel }}^2$, 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 $(A^{k}x{)}_{k\in \mathbb{N}}$, 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.