en تخصصي Special پژوهشي Research paper <br> <gdiv></gdiv> <div style="text-align: justify;"><span class="fontstyle0">Controlled frames in Hilbert spaces have been recently introduced by P. Balazs and etc. for improving the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper we develop a theory based on g-fusion frames on Hilbert spaces, which provides exactly the frameworks not only to model new frames on Hilbert spaces but also for deriving robust operators. In particular, we can define analysis, synthesis and frame operators with representation space compatible for (</span><span class="fontstyle2">C,C</span><span class="fontstyle3">&#39;</span><span class="fontstyle0">)-Controlled g-fusion frames, which even yield a reconstruction formula. Also, some useful concepts such as Q-dual and perturbation are introduced and investigated.</span></div> G-fusion frame, Controlled fusion frame, Controlled g-fusion frame, Q-dual. 179 191 http://ijmsi.ir/browse.php?a_code=A-10-4439-1&slc_lang=en&sid=1 Habib shakoory habibshakoory@yahoo.com 100319475328460010029 100319475328460010029 No Department of Mathematics, Shabestar Branch, Islamic Azad University Shabestar, Iran Reza ahmadi rahmadi@tabrizu.ac.ir 100319475328460010030 100319475328460010030 Yes Research Institute for Fundamental Sciences, University of Tabriz, Tabriz, Iran Gholamreza Rahimlou grahimlou@gmail.com 100319475328460010031 100319475328460010031 No Department of Mathematics, Faculty of Tabriz Branch, Technical and Vocational University (TVU), East Azarbaijan, Iran Vahid Sadri vahidsadri57@gmail.com 100319475328460010032 100319475328460010032 No Department of Mathematics, Faculty of Tabriz Branch, Technical and Vocational University (TVU), East Azarbaijan, Iran