Properties of bounded representations for G-frames

The purpose of the paper is to analyze g-frames of the form {φTi∈B(H,K)}i=0∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\varphi T^{i} \in B(\mathcal {H},\mathcal {K})\}_{i=0}^\infty $$\end{document}, where T∈B(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\in B(\mathcal {H})$$\end{document} and φ∈B(H,K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in B(\mathcal {H},\mathcal {K})$$\end{document}, and discuss the properties of the operator T. We consider stability of g-Riesz sequences of the form {φTi∈B(H,K)}i=0∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\varphi T^{i} \in B(\mathcal {H},\mathcal {K})\}_{i=0}^\infty $$\end{document}. Finally, a weighted representation of a g frame is introduced and some of its properties are presented. We provide a sufficient condition for a given g-frame Λ={Λi∈B(H,K)}i=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda =\{\Lambda _{i}\in {B(\mathcal {H},\mathcal {K})}\}_{i=1}^\infty $$\end{document} to be represented by an operator T∈B(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\in B(\mathcal {H})$$\end{document} and a sequence {ai}i=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{a_i\}_{i=1}^\infty $$\end{document}.