A Summary Of The Progress Of Elementary

In this paper, we shall be concerned with the J ordan elementary operator UA,B: X → AXB + BXA,(A, B ∈ A)de?ned on some normed algebra A. The compute of norm is one of the most important problems in theory of elementary operators. In the case of J ordan elementary operator,the value of c in the inequality ∥UA,B∥ ≥ c∥A∥∥B∥ is the mainly problem. We will give a survey of this problem.We mainly consider the work of M.M athieu, M.Barraa, B.M agajna, R.M.T imoney and A.Seddik, etc. The value of c may be23, 2(√2- 1) and 1 in di?erent cases. If E is an in?nite dimension Banach space, then c ≤ 1 for the J ordan elementary operator de?ned on B(E). It is an important method for the study of elementary operator to use some special operators such as the operator of ?nite rank and Hilbert- Schmidt operator. Another important method is matrix technique, many important results about elementary operator on in?nite dimension space can be induced by the case of space of two dimensional. We can also study elementary operator by using of completely bounded norm and Haagerup norm. We give some proofs of same important results and methods.