In　this　paper,　we　generalize　the　Roper-Suffridge　extension　operator　from　C-n　to　Banach　spaces.　It　is　proved　that　this　operator　preserves　the　biholomorphic　epsilon　starlikeness　on　some　domains　in　Banach　spaces.　From　these,　we　may　construct　a　lots　of　concrete　examples　about　biholomorphic　epsilonepsilon　starlike　mappings　on　some　domains　Omega　in　C-n,　or　Hilbert　spaces,　or　Banach　spaces　from　univalent　epsilon　starlike　functions　on　the　unit　disc　U　in　C.　Meanwhile,　the　growth　theorems　of　the　corresponding　mappings　are　given.　Some　results　of　Gong　and　Liu,　Roper　and　Suffridge,　Graham　et　al.　in　C-n　are　extended　to　Hilbert　spaces　or　Banach　spaces.　(C)　2004　Elsevier　Inc.　All　rights　reserved.