Research On Linear Preserving Mappings Which Are On B(H)

The study of operator algebra theory began in 30times of the 20th century. With the fast development of the theory, now it has become a hot branch playing the role of an initiator in modern mathematics. It has unexpected relations and interinfiltrations with quantum mechanics, noncommutative geometry, linear system and control theory, indeed number theory as well as some other important branches of mathematics. In order to discuss the structure of operator algebras, in recent years, many scholars both here and abroad have focused on linear preserving mappings on operator algebras and have introduced more and more new methods. The preserving problem on operator algebra is to research some mappings that preserve characterization on operator algebra. On the basis of existing papers, in this paper we mainly and detailedly discuss linear bounded surgections that preserve quasisimilarity, and elementary operators of length 1 and 2 that preserve unitary similarity.This paper contains three chapters:Chapter 1 mainly introduces some notations, definitions and some well-known theorems are given. Firstly, we give some notations. Subsequently, we introduce the definations of spectrum , similarity, quasisimilarity, unitary similarity, and elementary operators etc. Finally, we give some well-known theorems.In chapter 2, we discuss bounded linear mappings that preserve quasisimilarity. Let H be a separable infinite dimensional Hilbert space and B(H) the Banach algebra of all bounded linear operators on H. In [1],authors researched the linear mappings that are preserve similarity in infinite dimension. They charactered the bounded linear surjections on B(H) that preserve similarity in both directions by characterizing the similarity-invariant subspaces of B(H). In [2],author charactered bounded linear surjections on B(H) that preserve asymptopical similarity. In this chapter, let H be a separable infinite dimensional Hilbert space and B(H) the Banach algebra of all bounded linear operators on H. we study the quasisimilarity-invariant subspaces of B(H) and quasisimilarity-preserving linear maps on B(H). In this chapter, it is proved that the quasisimilarity-invariant subspaces of B(H) have three forms, that is {0}, CI and B(H) and that every bounded linear surjective mapping on B(H)which preserve quasisimilarity is nonzero scalar multiple of either an automorphism or an anti-automorphism by using the method of linear operator approximation.In chapter 3, We discuss elementary operators . In [3],author had characterised the elementary operators of length 1 that preserve similarity. We discuss the elementary operators on B^H). In the first place , we give the characterasitons for elementary operators of length 1 that preserve unitary similarity. Subsequently, let elementary operator A{X) = A\XBi -I- A2XB2 of length 2 on B(7i) preserves unitary similarity, if / € ran(A), then A(X) = auAX^A'1 for every X € B(H), and A = {AX,A2) € B{H?H,H),an e C. Similarly, if I e ran(A), then this elementary operator of length 2 preserving similarity (asymptopical similarity or quasisimilarity) if and only if A(X) = auAX^A'1 for every X € B(H), and A = (Au A2) € B(H e H, H), an € C.