Let H be a separable Hilbert space and let \(\{x_{n}\}\) be a sequence in H that does not contain any zero elements. We say that \(\{x_{n}\}\) is a Bessel-normalizable or frame-normalizable sequence if the normalized sequence \({\bigl \{\frac{x_n}{\Vert x_n\Vert }\bigr \}}\) 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 \(\bigl \{\frac{A^{n} x}{\Vert A^{n}x\Vert }\bigr \}_{n\ge 0,\, x\in S}\) associated with a normal operator \(A:H\rightarrow H\) 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 \(\bigl \{\frac{A^{n} x}{\Vert A^{n}x\Vert }\bigr \}_{n\ge 0,\, x\in S}\) is not a frame for H whenever \(\{A^{n}x\}_{n\ge 0,\,x\in S}\) is a frame for H.

令H为可分离希尔伯特空间，令\(\{x_{n}\}\)为H中不包含任何零元素的序列。如果归一化序列\ ({\bigl \{\frac{x_n}{\Vert x_n\Vert }\bigr \}}\)分别是贝塞尔序列或H的框架。本文证明了序列可帧归一化和不可帧归一化的几个充要条件。框架归一化序列的微扰定理也得到了证明。作为应用，我们证明巴拉兹-斯托耶娃猜想对于贝塞尔归一化序列成立。最后，我们应用我们的结果来部分回答 Aldroubi 等人提出的悬而未决的问题。迭代系统\(\bigl \{\frac{A^{n} x}{\Vert A^{n}x\Vert }\bigr \}_{n\ge 0,\, x\in S}\)与普通算子\(A:H\rightarrow H\)和H的可数子集S相关联，是H的框架。特别是，如果S是有限的，那么我们可以证明\(\bigl \{\frac{A^{n} x}{\Vert A^{n}x\Vert }\bigr \}_{n当 \(\{A^{n}x\}_{n\ge 0 ,\,x\in S}\) 是一个时，\ge 0,\, x\in S}\)就不是H的框架H的框架。