Frame-normalizable sequences

Let H be a separable Hilbert space and let \(\{x_{n}\}\) $\{x_n\}$ be a sequence in H that does not contain any zero elements. We say that \(\{x_{n}\}\) $\{x_n\}$ is a Bessel-normalizable or frame-normalizable sequence if the normalized sequence \({\bigl \{\frac{x_n}{\Vert x_n\Vert }\bigr \}}\) $\{\frac{x_n}{\Vert x_n\Vert }\}$ is a Bessel sequence or a frame for H, respectively. In this paper, several necessary and sufficient conditions for sequences to be frame-normalizable and not frame-normalizable are proved. Perturbation theorems for frame-normalizable sequences are also proved. As applications, we show that the Balazs–Stoeva conjecture holds for Bessel-normalizable sequences. Finally, we apply our results to partially answer the open question raised by Aldroubi et al. as to whether the iterative system \(\bigl \{\frac{A^{n} x}{\Vert A^{n}x\Vert }\bigr \}_{n\ge 0,\, x\in S}\) $\{\frac{A^{n}x}{\Vert A^{n}x\Vert }{\}}_{n\ge 0,x\in S}$ associated with a normal operator \(A:H\rightarrow H\) $A:H\to H$ and a countable subset S of H, is a frame for H. In particular, if S is finite, then we are able to show that \(\bigl \{\frac{A^{n} x}{\Vert A^{n}x\Vert }\bigr \}_{n\ge 0,\, x\in S}\) $\{\frac{A^{n}x}{\Vert A^{n}x\Vert }{\}}_{n\ge 0,x\in S}$ is not a frame for H whenever \(\{A^{n}x\}_{n\ge 0,\,x\in S}\) ${\{A^{n}x\}}_{n\ge 0,x\in S}$ is a frame for H.