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　Gavruta.　(C)　2012　Elsevier　Ltd.　All　rights　reserved.