Let \({\mathcal {H}}\) be a separable Hilbert space. It is known that the finite sum of Bessel sequences in \({\mathcal {H}}\) is still a Bessel sequence. But the finite sum of generalized notions of frames does not necessarily remain stable in its initial form. In this paper, for a prescribed Bessel sequence \(F=\{f_n\}_{n=1}^\infty \), we introduce and study \({\mathcal {KF}}\), the set consisting of all operators \(K\in {\mathcal {B}}({\mathcal {H}})\), such that \(\{f_n\}_{n=1}^\infty \) is a K-frame. We show that \({\mathcal {KF}}\) is a right ideal of \({\mathcal {B}}({\mathcal {H}})\). We indicate by an example that \({\mathcal {KF}}\) is not necessarily a left ideal. Moreover, we provide some sufficient conditions for the finite sum of K-frames to be a K-frame. We also use some examples to compare our results with existing ones. These examples demonstrate that our achievements do not depend on the available results. Furthermore, we study the same subject for K-g-frames and controlled frames and get some similar significant results.

令\({\mathcal {H}}\)是一个可分离的希尔伯特空间。已知\({\mathcal {H}}\)中贝塞尔序列的有限和仍然是贝塞尔序列。但是框架的广义概念的有限和不一定在其初始形式中保持稳定。在本文中，对于一个规定的贝塞尔序列\(F=\{f_n\}_{n=1}^\infty \)，我们引入并研究了\({\mathcal {KF}}\)，集合由所有运算符\(K\in {\mathcal {B}}({\mathcal {H}})\)，使得\(\{f_n\}_{n=1}^\infty \)是K -框架。我们证明\({\mathcal {KF}}\)是\({\mathcal {B}}({\mathcal {H}})\)的正确理想. 我们通过一个例子表明\({\mathcal {KF}}\)不一定是左理想。此外，我们提供了一些充分条件，使K帧的有限和成为K帧。我们还使用一些示例将我们的结果与现有结果进行比较。这些例子表明我们的成就不依赖于可用的结果。此外，我们针对K - g帧和受控帧研究了相同的主题，并得到了一些类似的显着结果。