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
Department of Mathematics, Faculty of Science, United Arab Emirates University, P.O. Box 17551, Al Ain, United Arab Emirates
Received 3 November 2000
Copyright © 2001 B. P. Duggal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.