Open Access

# $pg$-frame in Banach spaces

### Abstract

For extending the concepts of $p$-frame, frame for Banach spaces and atomic decomposition, we will define the concept of $pg$-frame and $g$-frame for Banach spaces, by which each $f\in X$ ($X$ is a Banach space) can be represented by an unconditionally convergent series $f=\sum g_{i}\Lambda_{i},$ where $\{\Lambda_{i}\}_{i\in J}$ is a $pg$-frame, $\{g_{i}\}\in(\sum\oplus Y_{i}^{*})_{l_q}$ and $\frac{1}{p}+\frac{1}{q}=1$. In fact, a $pg$-frame $\{\Lambda_{i}\}$ is a kind of an overcomplete basis for $X^{*}.$ We also show that every separable Banach space $X$ has a $g$-Banach frame with bounds equal to $1.$

### Article Information

 Title $pg$-frame in Banach spaces Source Methods Funct. Anal. Topology, Vol. 13 (2007), no. 3, 201-210 MathSciNet MR2356754 Copyright The Author(s) 2007 (CC BY-SA)

### Authors Information

M. R. Abdollahpour
Department of mathematics, Tabriz university, Tabriz, Iran

M. H. Faroughi
Department of mathematics, Tabriz university, Tabriz, Iran

A. Rahimi
Department of mathematics, Tabriz university, Tabriz, Iran

### Citation Example

M. R. Abdollahpour, M. H. Faroughi, and A. Rahimi, $pg$-frame in Banach spaces, Methods Funct. Anal. Topology 13 (2007), no. 3, 201-210.

### BibTex

@article {MFAT365,
AUTHOR = {Abdollahpour, M. R. and Faroughi, M. H. and Rahimi, A.},
TITLE = {$pg$-frame in Banach spaces},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {13},
YEAR = {2007},
NUMBER = {3},
PAGES = {201-210},
ISSN = {1029-3531},
MRNUMBER = {MR2356754},
URL = {http://mfat.imath.kiev.ua/article/?id=365},
}