## Approximative $K$-Atomic Decompositions and frames in Banach Spaces. (arXiv:1901.05950v1 [math.FA])

[L. Gavruta, Frames for Operators, Appl. comput. Harmon. Anal. 32(2012),
139-144] introduced a special kind of frames, named $K$-frames, where $K$ is an
operator, in Hilbert spaces, 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 $\mathcal{X}_d$-frame and approximative
$\mathcal{X}_d$-Bessel sequence are introduced and studied. Two necessary
conditions are given under which an approximative $\mathcal{X}_d$-Bessel
sequence and approximative $\mathcal{X}_d$-frame give rise to a bounded
operator with respect to which there is an approximati查看全文