In　this　paper,　we　give　some　equivalent　conditions　on　a　Banach　frame　for　a　Banach　space　by　using　the　pseudoinverse　operator.　We　also　consider　the　stability　of　a　Banach　frame　for　a　Banach　space　X　with　respect　to　Xd　or　an　Xd-frame　for　a　Banach　space　X　under　perturbation.　These　results　generalize　and　improve　the　related　works　of　Balan,　Casazza,　Christensen,　Stoeva　and　Jian　et　al.　©　2010　Institute　of　Mathematics,　Academy　of　Mathematics　and　Systems　Science,　Chinese　Academy　of　Sciences,　Chinese　Mathematical　Society　and　Springer-Verlag　Berlin　Heidelberg.