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International Journal of Mathematics and Mathematical Sciences
Volume 27, Issue 9, Pages 573-582
http://dx.doi.org/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 Hindawi Publishing Corporation. 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 (H) denote the algebra of operators on a Hilbert space H into itself. Let d=δ or Δ, where δAB:(H)(H) is the generalized derivation δAB(S)=ASSB and ΔAB:(H)(H) is the elementary operator ΔAB(S)=ASBS. Given A,B,S(H), we say that the pair (A,B) has the property PF(d(S)) if dAB(S)=0 implies dAB(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.