  Recent research has shown that the properties of overcomplete Gabor frames
and frames arising from shift-invariant systems form a precise match with
certain conditions that are necessary for a frame in $L^2(\mathbf R)$ to have a
representation $\{T^k \varphi\}_{k=0}^\infty$ for some bounded linear operator
$T$ on $L^2(\mathbf R)$ and some $\varphi \in L^2(\mathbf R).$ However, for
frames of this type the existence of such a representation has only been
confirmed in the case of Riesz bases. This leads to several open questions
connecting dynamical sampling, coherent states, frame theory, and operator
theory. The key questions can either be considered in the general functional
analytic context of operators on a Hilbert space, or in the specific situation
of Gabor frames in $L^2(\mathbf R).$
