Elementary Operator Norm And Related Issues

The study of operator algebra theory began in 20th century. Since it is usedwidely in mathematics and other scientific branches, it got great development atthe beginning of the 20th century. Elementary operators are important linear map-pings. In recent years, many scholars both here and abroad have focused on manycharacterization on elementary operators. In this paper we mainly and detailedlydiscuss the norm of elementary operators, norm attainability of some elementaryoperators and the density of some certain sets about the norm of some elementaryoperators.This paper contains three chapters:Chapter 1 mainly introduces some notations, definitions and some well-knowntheorems. Firstly, we give some notations. Subsequently, we introduce the defini-tions of elementary operators, numerical range, normal maximal numerical range,spectrum, norm attainability etc. Finally, we give some well-known theorems suchas polar decomposition theorem and spectral decomposition theorem.In chapter 2, we discuss the norm of some certain elementary operators. Let Hbe a separable infinite dimensional Hilbert space and B(H) be the Banach algebra ofall bounded linear operators on H. In [1], J. Stampfii compute the norm of generalderivation. In [2], M. Barraa and M. Boumazgour find the equality condition for‖△A,B‖=‖A‖‖B‖+ 1. In this chapter, firstly, we proved that‖UA,B‖=2‖A‖‖B‖if and only if‖A*B‖=‖A‖‖B‖and WN(B*A)∩WN(A,B)≠(?)(A,b≠0) and givesome other sufficient or necessary condition for‖UA,B‖= 2‖A‖‖B‖.Secondly,wegive some sufficient condition for 0∈WB(A*B)∪WA(B*A)when‖UA,B‖=‖A‖‖B‖,and we proved that if‖B*A‖∈WA(B*A),‖AB*‖∈WB*(AB*)and‖A‖‖B‖∈WB*(A*B)∩WA*(BA*),then‖UA,B‖= [(‖A‖‖B‖+‖AB*‖)(‖A‖‖B‖+‖B*A‖)](?)Lastly, an example is given to show that‖A*B‖=‖A‖‖B‖is necessary but notsufficient condition for‖UA,B‖= 2‖A‖‖B‖, so we negatively answer the questionsposed by A.Seddik in [3].In chapter 3, we discuss some characterization on elementary operators△A,B,where△A,B(X) = AXB+X. Firstly, we discuss the norm attainability of elementaryoperator△A,B. Secondly, we discuss the density of two sets {(A, B) :‖△A,B‖< 1+‖A‖‖B‖+‖}and{(A,B):‖UA,B‖<2‖A‖‖B‖}in B(H).Lastly,we mainlyproved that if A and B are paranormal operators,(?)λA∈isoó(△AB)andλ≠1,then Ho(△AB-λ)=(△AB-λ)～(-1)(0).