A Putnam–Fuglede commutativity property for Hilbert space operators

Given Hilbert space operators A,B∈B(H)A,B∈B(H), define δA,BδA,B and △A,B△A,B in B(B(H))B(B(H)) by δA,B(X)=AX−XBδA,B(X)=AX−XB and △A,B(X)=AXB−X△A,B(X)=AXB−X for each X∈B(H)X∈B(H). An operator A∈B(H)A∈B(H) satisfies the Putnam–Fuglede properties δ  , respectively △ (notation: A∈PF(δ)A∈PF(δ), respectively A∈PF(△)A∈PF(△)), if for every isometry V∈B(H)V∈B(H) for which the equation δA,V⁎(X)=0δA,V⁎(X)=0, respectively △A,V⁎(X)=0△A,V⁎(X)=0, has a non-trivial solution X∈B(H)X∈B(H), the solution X   also satisfies δA⁎,V(X)=0δA⁎,V(X)=0, respectively △A⁎,V(X)=0△A⁎,V(X)=0. We prove that an operator A∈B(H)A∈B(H) is in PF(△)PF(△) if and only if it is in PF(δ)PF(δ).