on fuglede–putnam theorem

for operators a and b, let com(a,b) stand for the set of operators x such that ax = xb. a pair (a,b) is said to have the (fp)-property if com(a,b) í com(a*,b*). let ˜ c denote the aluthge transform of a bounded linear operator c, (i) if a and b are invertible operators and (a,b) has the (fp)-property, then so is (˜a,˜b); (ii) if a and b are invertible and u ± ii and v ± ii are all invertible and (˜a,˜b) has the (fp)-property, then so is (a,b); (iii) if u²|a|u has the (fp)-property and a³ = i, then a is a unitary operator.