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Let  = || and  = || be the polar decompositions of  and  and let Com(, ) stand for the set of operators  such that  =  . A pair (, ) is said to have the FP-property if Com(, )  Com(*, *). Let  denote the Aluthge transform of a bounded linear operator . We show that (i) if  and  are invertible and (, ) has the FP-property, then so is  ; (ii) if  and  are invertible, the spectrums of both  and  are contained in some open semicircle and  has the FP-property, then so is (, ); (iii) if (, ) has the FP-property, then Com(, )  Com, moreover, if  is invertible, then Com(, ) = Com. Finally, if  and  and  is an operator such that *  = , then we prove that  for any .

Fuglede–Putnam type theorems via the Aluthge transform