Some Results About the Operator Perturbation of a K-Frame

In this paper we mainly study the stabilities of K-frames under the operator perturbation. Firstly, we provide several sufficient conditions of the operator perturbation for a K-frame by using a bounded linear operator T from H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_1}$$\end{document} to H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_2}$$\end{document}. We also give an equivalent characterization of the operator perturbation for a tight K-frame. Meanwhile, we correct two results which were obtained by Ramu. Lastly, we show that a K-frame can construct a T-frame by the perturbation of a bounded linear operator T. Our results generalize the remarkable results of the operator perturbation for a frame which were obtained by Casazza, Christensen, etc. when we take K=I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K = I$$\end{document}.