Approximately Numerical Range-preserving Maps On M2

Numerical range of operators is a very important concept and is extensively studied in both theory and applications, a lot of work have been done on maps preserving nu-merical range of operator and maps preserving numerical range of products of operators. And then, in this paper we mainly study the maps which almost preserves the numerical range. Let Mn be the algebra of n×n complex matrices. We say Φ:Mnâ†'Mn is a linear map which almost preserves the numerical range, if there is a small δ>0, such that ω(A)≤1hold for each A∈Mn, with W(Φ(A))(?)W(A)+δ, W(A)(?)W(Φ(A))+δ We call that Φ is a linear map which almost preserves the numerical range of products, for any ω(A)≤1, ω(B)≤1, A,B∈Mn, such that W(Φ(AB))(?)W(AB)+δ, W(AB)(?)W(Φ(AB))+δ hold.2×2matrices have unique properties in matrix theory. At the start of the study that a map almost preserves the numerical range, in this article, we characterize approx-imately numerical range-preserving maps on M2and the maps which almost preserves the numerical range of products of matrices, and also we show that such maps is a small perturbation of an automorphism or anti-automorphism.