Elementary Operator Norm

Operator theory is a very and important research field in the functional anal-ysis. Ever since von Neumann, Hilbert etc. established the operator theory at the beginning of 20 centuries, the operator theory has already quickly got to develop and seep through mathematics of each branch and become a through long not wane the research topic. Its research contents involves to many branches of the pure mathe-matics and the applied mathematics such as geometry theory, operator perturbation theory, matrix analysis, approximation theory, noncommutative probability theory, and many others. The study of elementary operators is an important branch in operator algebra theory, while the study of the norms of elementary operators are of great value, so it attracted the attention of many authors. In this paper we mainly discuss the norm of elementary operatorΔA,B and the norm equalities between the elementary operators with length 2 and the unitarily invariant norm inequalities and the singular value inequalities of the elementary operators with length 2.This paper contains three chapters:In Chapter 1, mainly introduces some notations, definitions. Firstly, we give some notations. Subsequently, we introduce the definitions of elementary operators, numerical range, the unitarily invariant norm etc.In Chapter 2, we dicuss the norm of elementary operatorΔA,B·Let H be a complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. Firstiy, we shall give necessary and sufficient conditions for any pair of operators A and B to satisfy the equation‖I-ΔA,B‖=1+‖A‖‖B‖+‖B‖, where I denote the identity operator on H andΔA,B(X)=AXB+XB ((?)X∈B(H)). Secondly, one sufficient and necessary conditions for‖MA;B+MC,D‖=‖A‖‖B‖+‖C‖‖D‖is obtained.In Chapter 3, we dicuss the unitarily invariant norm inequalities and the singu-lar value inequalities of the elementary operators with length 2. Let H be a separable infinite dimensional complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. In this chapter, we firstly discuss if X∈B0(H) and A, B∈B(H) such that one is positive and the other commutes with X, then |||AXB-BXA|||≤‖A‖‖B‖|||X||| for every unitarily invariant norm|||·||| and the usual operator norm‖·‖:Moreover, we prove several singular value inequalities for the elementary operators with length 2.