Quasinormality and Fuglede-Putnam theorem for (s,p)-w-hyponormal operators

We investigate several properties of Aluthge transform T-s = /T/ sU/T/ s of an operator T = U/T/. We prove (i) if T is (s, p)-w-hyponormal operator and Ts is quasinormal (resp., normal), then T is quasinormal (resp., normal), (ii) if T is (s, p)-w-hyponormal operator and Ts is a partial isometry, then T is quasinormal partial isometry, (iii) if T and T* are (s, p)-w-hyponormal operator, then T is normal, and (iv) FugledePutnam type theorem holds for a class p-w-hyponormal operator T with 0 < p = 1 if T satisfies a kernel condition ker (T). ker (T*).