Given Hilbert space operators \(A, B\in \mathcal {L}(H)\) . The pair (A, B) satisfies the Fuglede–Putnam–Aluthge property if \(AT=TB\) and \(T\in \mathcal {L}(H)\) implies \({\tilde{A}}T=T{\tilde{B}}\) , where \({\tilde{A}}\) is the Aluthge transform of A. We prove that the class of pairs (A, B) that possess the Fuglede–Putnam–Aluthge property includes pairs of quasinormal operators, pairs of partial isometries with a normal square, the class of pairs (A, B) such that A and \(B^*\) are p-hyponormal or log-hyponormal, pairs (A, B) where A is a dominant operator and \(B^*\) is p-hyponormal or log-hyponormal, the class of pairs (A, B) for which A and \(B^*\) are \(w_*\) -hyponormal and all pairs of operators satisfying the Fuglede–Putnam property (FP-property). We also show that, (1) If A is invertible, then (A, B) has the FP-property implies that \(({\tilde{A}}, {\tilde{B}})\) has the FP-property. (2) If A and B are iw-hyponormal, then (A, B) has the FP-property if and only if \(({\tilde{A}}, {\tilde{B}})\) has it too. We give some classes of operators A and B for which \(({\tilde{A}}, {\tilde{B}})\) has the FP-property.
