A new characterization of Anderson's Inequality inC_1-classes

摘要

Let Cal H be a separable infinite dimensional complex Hilbert space, and let Cal L(H) denote the algebra of all bounded linear operators on Cal H into itself. Let A=(A_1,A_2,dots,A_n), B=(B_1,B_2,dots,B_n) be n-tuples of operators in Cal L(H); we define the elementary operators DeltaA,B Cal L(H)mapstoCal L(H) by DeltaA,B(X)sumi=1^nA_iXB_i-X.In this paper, we characterize the class of pairs of operators A,BinCal L(H) satisfying Putnam-Fuglede's property, i.e, the class of pairs of operators A,BinCal L(H) such that sumi=1^nB_iTA_i= T implies sumi=1^nA_i^*TB_i^*=T for all TinCal C_1Cal(H) (trace class operators). The main result is the equivalence between this property and the fact that the ultraweak closure of the range of the elementary operator DeltaA,B is closed under taking adjoints. This leads us to give a new characterization of the orthogonality (in the sense of Birkhoff) of the range of an elementary operator and its kernel in C_1 classes.