| The study of operator algebra theory began in 1930s. With thefast development of the theory, now it has become a popular branch playing the roleof an initiator in modern mathematics. It has unexpected relations and infiltrationwith quantum mechanics, differential geometry, linear system and control theory,indeed number theory as well as some other important branches of mathematics.Elementary operators are important linear mapping. In recent years, many scholarshave focused on many characterization on elementary operators. On the basis of ex-isting papers, in this paper we mainly and detailedly discuss the norm of elementaryoperators, subnormality of some elementary operators and the maximal numericalrange of some elementary operators.This paper contains three chapters:In Chapter 1, we introduce some notations, definitions and some well-knowntheorems that will be used in last two chapters. Firstly, we give some notations.Subsequently, we introduce the definitions of elementary operators, numerical range,maximal numerical range, spectrum, approximate point spectrum and subnormaloperator etc. Finally, we give some well-known theorems.In Chapter 2, we discuss the norm of some elementary operators in B(H) ofall bounded linear operators acting on a complex separable infinite dimensionalHilbert space H. Firstly, we prove that‖△‖=‖A‖‖B‖+‖C‖if and only if‖A*C‖=‖A‖‖C‖and WN(A*C)∩WN(B)≠φ(A,B,C≠0) and find a lowerbound of△, that is,‖△‖≥(?)‖‖B‖A+(?)C‖. Secondly, we find a lower bound ofε, that is, if A1≠0, B1≠0, (λ1,λ2,…,λn-1)∈WA1(A2*A1,A3*A1,…,An*A1)N, and(μ1,μ2,…,μn-1)∈WB1(B2*B1, B3*B1,…,Bn*B1)N, then‖ε‖≥sup{|‖A1‖‖B1‖+λ1μ1‖A2‖‖B2‖+…+λn-1μn-1‖An‖‖Bn‖|}, also we prove that if‖ε‖=‖A1‖‖B1‖+‖A2‖‖B2‖+…+‖An‖‖Bn‖, then WA1(A2*A1,…,An*A1)N∩(?)≠φ. Where A1≠0, B1≠0.In Chapter 3, we discuss other characterization on elementary operators inB2(H). Where B2(H) denotes Hilbert-Schmidt class. We prove that if AC=CAand△is subnormal, then (?)A+C is a subnormal operator. Whereλ∈σap(B*).Also we research the maximal numerical range of some elementary operators. |
