In　this　paper,　we　generalize　the　Roper-Suffridge　extension　operator　from　Cn　to　Banach　spaces.　It　is　proved　that　this　operator　preserves　the　biholomorphic　ε　starlikeness　on　some　domains　in　Banach　spaces.　From　these,　we　may　construct　a　lots　of　concrete　examples　about　biholomorphic　ε　starlike　mappings　on　some　domains　Ω　in　Cn,　or　Hilbert　spaces,　or　Banach　spaces　from　univalent　ε　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　Cn　are　extended　to　Hilbert　spaces　or　Banach　spaces.　©　2004　Elsevier　Inc.　All　rights　reserved.