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

Abstract

Letℬ(H)denote the algebra of operators on a Hilbert spaceHinto itself. Letd=δorΔ, whereδAB:ℬ(H)→ℬ(H)is the generalized derivationδAB(S)=AS−SBandΔAB:ℬ(H)→ℬ(H)is the elementary operatorΔAB(S)=ASB−S. GivenA,B,S∈ℬ(H), we say that the pair(A,B)has the propertyPF(d(S))ifdAB(S)=0impliesdA∗B∗(S)=0. This paper characterizes operatorsA,B, andSfor which the pair(A,B)has propertyPF(d(S)), and establishes a relationship between thePF(d(S))-property of the pair(A,B)and the range-kernel orthogonality of the operatordAB.