This paper investigates a fundamental problem in frame theory: characterizing frames of the form \(\{T^k \varphi \}_{k=0}^\infty \) , where T is a bounded linear operator on a Hilbert space \(\mathcal {H}\) and \(\varphi \in \mathcal {H}\) . Our primary goal is to identify conditions under which such an iterative sequence forms a frame for \(\mathcal {H}\) . In particular, we provide a complete characterization of diagonal operators on \(\ell ^2(\mathbb {N})\) that generate frames of this form. Additionally, we establish necessary conditions for a frame \(\{f_k\}_{k=0}^\infty \) to admit a representation as \(\{T^k f_1\}_{k=0}^\infty \) . Further related results are also presented.
