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

Abstract

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. One of the last extensions of frames is weighted and controlled frames that introduced by P.Balazs, J-P. Antoine and A. Grybos 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 gframes will be defined and we will show that controlled g-frames are equivalent to g-frames and so the controlled operators C and C 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.

Weaving continuous controlled K-g-fusion frames in Hilbert spaces

“…P. Balaz et al [3] introduced controlled frame to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In recent times, several generalizations of controlled frame namely, controlled K-frame [22], controlled g-frame [23], controlled fusion frame [19], controlled g-fusion frame [28], controlled K-g-fusion frame [24] etc. have been appeared.…”

Some Results on Controlled K−Frames in Hilbert Spaces

“…Controlled frames extended to g-frames in [17] and for fusion frames in [15]. In this section, the concept of controlled frames and controlled Bessel sequences will be extended to K-frames and it will be shown that controlled K-frames are equivalent K-frames.…”

“…The main advantage of these frames lies in the fact that they retain all the advantages of standard frames but additionally they give a generalized way to check the frame condition while offering a numerical advantage in the sense of preconditioning. Recent developments in this direction can be found in [14][15][16][17][18] and the references therein.…”

On controlled frames in Hilbert C∗-modules

“…Basic properties and some applications of this operator for Bessel sequences, frames and Riesz basis have been proved by Peter Balazs in his Ph.D habilation [3]. Recently, the concept of multipliers extended and introduced for continuous frames [6], fusion frames [2], p-Bessel sequences [26], generalized frames [25], controlled frames [27], Banach frames [12,13], Hilbert C * -modules [23] and etc. The symbol of m has important role in the studying of multiplier operators.…”