Generalized Anderson’s Inequality

Let H be a separable infinite-dimensional complex Hilbert space and let BŽH. denote the algebra of operators on H into itself. We study the elementary operator : BŽH. BŽH. defined by ŽX. AXB CXD, where A and C Žrespec- tively, B and D. are nonzero normal commuting operators. We prove that Ži. ŽX. S S for all S NŽ . Žthe kernel of . and for all X BŽH.or Žii. ŽX. S p S pfor all S NŽ . Cp Žthe von Neumann Schat- ten class., 1 p , p 2, and for all X BŽH. such that Ž X. Cp if and only if NŽA. NŽC. NŽB*. NŽD*. 04.