Some equalities and inequalities for g-Bessel sequences in Hilbert spaces (II)

Abstract

G-frames, which include many generalizations of frames such as frames of subspaces or fusion frames, oblique frames, and pseudo-frames, are natural generalizations of frames in Hilbert spaces. They have some properties similar to those of frames in Hilbert spaces, but not all of their properties are similar. For example, exact frames are equivalent to Riesz bases, but exact g-frames are not equivalent to g-Riesz bases. Some authors have extended the equalities and inequalities for frames and dual frames to g-frames and dual g-frames in Hilbert spaces. In this paper, we establish some new equalities and inequalities for g-Bessel sequences or g-frames in Hilbert spaces. We also give a necessary and sufficient condition that the equality occurs in one of these inequalities. Our results generalize and improve the remarkable results which had been obtained by Balan, Casazza and Gavruta.