  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.
