(adsbygoogle = window.adsbygoogle || []).push({}); 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 = T_{0}implies T = T_{0}.

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,T_{0}x>, 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> = <T_{0}v,x>

<v,Tx> (This one only has one T on the inner product).

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# Homework Help: Linear operator problems

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