The Banach spaces associated with g-frames

In this paper, we introduce the Banach spaces induced by a g-frame and lp(⊕i∈NHi),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^p(\oplus _{i\in {\mathbb{N}}} {H_i}),$$\end{document} where 1≤p<2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p<2.$$\end{document} We study the different aspects of these spaces corresponding to reconstructions, existence and dilations. Specially, we obtain that for all closed subspaces of a Hilbert space H,  only the finite dimensional ones with a g-orthonormal basis can be realized as such a Banach space associated a g-frame. We also prove that under some conditions of the g-frame, the g-frame expansion of any element in the Banach space associated with it converges in both the Hilbert space norm and the associated Banach norm. Moreover, we give a dilation result of such space with the dilation properties of g-frames.