Executive Summary : | The theoretical research of frames for Banach spaces is quite different from that of Hilbert spaces. Due to the lack of an inner product, frames for Banach spaces were simply defined as sequence of linear functionals in , the dual of , rather than a sequence of basis-like elements in itself. However, semi-inner products for Banach spaces make possible the development of inner product type arguments in Banach spaces. The concept of a family of local atoms in a Banach space with respect to a BK-space was introduced by Dastourian and Janfada using a semi-inner product. This concept was generalized to an atomic system for a bounded operator , called atomic system and it has been led to the definition of a new frame with respect to the operator , called - frame. It is proposed to analyze concepts for Hilbert frames to - -frames in Banach spaces. A classical result says that a sufficiently small perturbation of an orthonormal basis gives a Riesz basis. Motivated by the theory of perturbation of Hilbert frames, it is proposed to find results about perturbations of “atomic systems'' for operators in Banach spaces. Recent studies in K-frames indicate that the scalable K-frames and -scalable K-frames can be used to manage the data loss in the domain of signal communication. Motivated from this idea, researchers are interested in the construction which modifies a given K-frame into a Parseval K-frame or a tight K-frame. Characterizations of K-frames that can be scaled to a Parseval K-frames or tight K-frames can be explored. One can study the scalability of K-frames, that is the existence of scaling sequence, as a consequence of the scaled sequence, is called a Parseval K-frame. It will be quiet interesting if we can analyse the uniqueness of the scalability of K-frames. |