Approximative K-atomic decompositions and frames in Banach spaces. Issue 1 (5th April 2018)

• Record Type: Journal Article
• Authors: Jahan, Shah
• Is Part Of: Arab journal of mathematical sciences. Volume 26:Issue 1/2(2020)
• Page Start: 153
• Page End: 166
• Publication Date: 2018-04-05
• Subjects: Frames -- K-frames -- Atomic decomposition -- K-atomic decomposition -- d-Bessel sequence -- d-frames; 42A38 -- 46B15 -- 42C15 -- 42C30
• DOI: 10.1016/j.ajmsc.2019.03.003
• Languages: English
• ISSNs: 1319-5166

Abstract

Abstract : L. Gǎvruţa (2012) introduced a special kind of frames, named K -frames, where K is an operator, in Hilbert spaces, which is significant in frame theory and has many applications. In this paper, first of all, we have introduced the notion of approximative K -atomic decomposition in Banach spaces. We gave two characterizations regarding the existence of approximative K -atomic decompositions in Banach spaces. Also some results on the existence of approximative K -atomic decompositions are obtained. We discuss several methods to construct approximative K -atomic decomposition for Banach Spaces. Further, approximative  d -frame and approximative  d -Bessel sequence are introduced and studied. Two necessary conditions are given under which an approximative  d -Bessel sequence and approximative  d -frame give rise to a bounded operator with respect to which there is an approximative K -atomic decomposition. Example and counter example are provided to support our concept. Finally, a possible application is given.