Power Stability of Finitely Generated Dynamical Frames for Positive Operators

Let $\mathrm{\Lambda }=\{{\lambda }_j{\}}_{j\in \mathbb{N}}\subset [0,1)$ be a finite union of Carleson sequences, and let ${\mathrm{\Lambda }}^p=\{{\lambda }_j^p{\}}_{j\in \mathbb{N}}$ for $p>0$. We prove that $\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}^p$ admit compatible Hartmann-type cluster decompositions and that, under the natural coordinate identification, $H^2(\mathrm{\Lambda })=H^2({\mathrm{\Lambda }}^p)\mathrm{.}$ After taking account of the change in the normalization weights, this trace-space identity gives ( \operatorname{Ran}(T_\Lambda) = \operatorname{Ran}(T_{\Lambda^p}) ) for the associated weighted evaluation operators. The characterization of finitely generated dynamical frames for normal operators then implies that, for every positive operator $A\in \mathcal{B}(H)$, every finite index set $I$, and every $p>0$, $\{A^n{f}_i{\}}_{n\in \mathbb{N},i\in I}\text{ is a frame for }H$ if and only if $\{A^{pn}{f}_i{\}}_{n\in \mathbb{N},i\in I}\text{ is a frame for }H\mathrm{.}$ In particular, for every $q\in {\mathbb{N}}^{+}$, the subfamily $\{A^{qn}{f}_i{\}}_{n\in \mathbb{N},i\in I}$ is still a frame, and hence every such dynamical frame has infinite excess. An example shows that the positivity assumption cannot in general be replaced by normality.