Power Stability of Finitely Generated Dynamical Frames for Positive Operators

Abstract

Let \( \Lambda=\{\lambda_j\}_{j\in\mathbb N}\subset[0,1) \) be a finite union of Carleson sequences, and let \( \Lambda^p=\{\lambda_j^p\}_{j\in\mathbb N} \) for \(p>0\). We prove that \(\Lambda\) and \(\Lambda^p\) admit compatible Hartmann-type cluster decompositions and that, under the natural coordinate identification, \( H^2(\Lambda)=H^2(\Lambda^p). \) 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 \(p>0\), \( \{A^nf_i\}_{n\in\mathbb N,\,i\in I} \text{ is a frame for }H\) if and only if \( \{A^{pn}f_i\}_{n\in\mathbb N,\,i\in I} \text{ is a frame for }H. \) 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.