Abstract: In this paper, some characterizations of isometric operators and ε-isometric opertors from 2-dim Banach Space E₂ into L¹[0, 1] are given, where E₂ denotes a Banach Space whose unit ball is the union of a polygon C and the region enclosd by C in the plane. Also,the answer for IAP(Isometric Approximation Problem) on B(E₂,L¹[0, 1]) is obtained.

Key words: Isometric operators, Isometric approxiamation, Almost isometry