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.
In this paper, a new concept related to the frame theory is introduced; the notion of pair frame. By investigating some properties of such frames, it is shown that pair frames are a generalization of ordinary frames. Some classes of pair frames are considered such as (p, q)-pair frames and near identity pair frames.2000 Mathematics Subject Classification. Primary 42C15.
The concept of (p, q)-pair frames is generalized to (ℓ, ℓ * )-pair frames. Adjoint (conjugate) of a pair frames for dual space of a Banach space is introduced and some conditions for the existence of adjoint (conjugate) of pair frames are presented.
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