1. The problem statement, all variables and given/known data 1. Let T be a linear operator on an inner product space V. Let U = TT*. Prove that U = TU*. 2. For a linear operator T on an inner product space V, prove that T*T = T0implies T = T0. 2. Relevant equations 3. The attempt at a solution 1. This appeared at the outset to be a relatively simple one, but for some reason I can't prove it: Let [tex]v,x \in V[/tex]. I know that U* = U, and <v,Ux> = <v,TT*x> = <T*v,T*x> <v,TU*x> = <T*v,U*x> =<T*v,TT*x> So there's always an extra T I can't get rid of. How do I prove this? 2. I'll try to prove <v,T*Tx> = <v,T0x>, knowing that (T*T)* = T*T But as before this leads nowhere because there is again an extra T I can't get rid off: <v,T*Tx> = <Tv,Tx> = <T0v,x> <v,Tx> (This one only has one T on the inner product).