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

G-frames have some properties similar to those of frames in complex Hilbert spaces, but not all of their properties are similar. Some authors have extended the equalities and inequalities for frames and dual frames to g-frames and dual g-frames in Hilbert spaces, respectively. In this work, we give some new equalities and inequalities for g-Bessel sequences in Hilbert spaces with pseudo-inverse operators. Our results generalize and improve the remarkable results which have been obtained by Balan, Casazza and Găvruţa.