International Journal of Mathematics and Mathematical Sciences 
Volume 27 (2001), Issue 9, Pages 573-582
doi:10.1155/S0161171201006159

Putnam-Fuglede theorem and the range-kernel orthogonality of derivations

B. P. Duggal

 Department of Mathematics, Faculty of Science, United Arab Emirates University, P.O. Box 17551, Al Ain, United Arab Emirates
Received 3 November 2000

Abstract

Let ℬ(H) denote the algebra of operators on a Hilbert space H into itself. Let d=δ or Δ, where δAB:ℬ(H)→ℬ(H) is the generalized derivation δAB(S)=AS−SB and ΔAB:ℬ(H)→ℬ(H) is the elementary operator ΔAB(S)=ASB−S. Given A,B,S∈ℬ(H), we say that the pair (A,B) has the property PF(d(S)) if dAB(S)=0 implies dA∗B∗(S)=0. This paper characterizes operators A,B, and S for which the pair (A,B) has property PF(d(S)), and establishes a relationship between the PF(d(S))-property of the pair (A,B) and the range-kernel orthogonality of the operator dAB.