【摘要】 设A=(A1,A2,…,An),B=(B1,B2,…,Bn),其中Ai,Bi∈B(H),i=1,2,…,n,定义初等算子RA,B:B(H)→B(H),MA1,B1:B(H)→B(H),分别为RA,B(X)=∑ni=1AiXBi,MA1,B1(X)=A1XB1,X∈B(H).记d(RA,B)为RA,B作用在H上所有的单位一秩算子的范数的上确界.利用d(RA,B)=∑ni=1‖Ai‖‖Bi‖成立的充要条件及正规代数数值域的定义,研究了d(RA,B)的一些性质,给出了n=2时d(RA,B)=‖A1‖‖B1‖+‖A2‖‖B2‖成立的新的充要条件并且估计了d(MA1,B1+MI,B2)的下界.更多还原

【Abstract】 For two n-tuples A=(A1,A2,…,An),B=(B1,B2,…,Bn),Ai,Bi∈B(H),i=1,2,…,n,the elementary operators RA,B:B(H)→B(H) and MA1,B1:B(H)→ B(H) are defined respectively by RA,B(X)=∑ni=1AiXBi,MA1,B1(X)=A1XB1,X∈B(H).d(RA,B) is denoted by the supremum of the norm of RA,B over all unit rank one operators on H.According to the sufficient and necessary conditions for d(RA,B)=∑ni=1‖Ai‖‖Bi‖ and the definition of the normalized algebraic numerical range,some properties of d(RA,B) are studied.And a new sufficient and necessary condition for d(RA,B)=‖A1‖‖B1‖+‖A2‖‖B2‖ for n=2 and the lower bound of d(MA1,B1+ MI,B2) are given.更多还原