Norms of elementary operators on rank-one operators
Guoxing Ji · Basic Sciences Journal of Textile Universities · 2007
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.