Controlled G-Frames and Their G-Multipliers in Hilbert spaces

Multipliers have been recently introduced by P. Balazs as operators for Bessel sequences and frames in Hilbert spaces. These are operators that combine (frame-like) analysis, a multiplication with a fixed sequence (called the symbol) and synthesis. Weighted and controlled frames have been introduced to improve the numerical efficiency of iterative algorithms for inverting the frame operator Also g-frames are the most popular generalization of frames that include almost all of the frame extensions. In this manuscript the concept of the controlled g-frames will be defined and we will show that controlled g-frames are equivalent to g-frames and so the controlled operators C and C0 can be used as preconditions in applications. Also the multiplier operator for this family of operators will be introduced and some of its properties will be shown.