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Homework Statement
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.
Homework Equations
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).