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Operator is normal iff ||Tv||=||T*v||, simple proof help

  1. Nov 22, 2009 #1
    http://img690.imageshack.us/img690/8659/linearalg.jpg [Broken]


    I am having trouble understanding 7.6 . Specifically, getting from the 2nd line to the first line. How do we know that T*T-TT* is self adjoint?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Nov 22, 2009 #2

    HallsofIvy

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    (AB)*= B*A* and (A*)*= A.

    The adjoint of T*T-TT* is (T*T- TT*)*= (T*T)*- (TT*)*= (T*)(T**)- (T**)T*. And the adjoint is 'dual'- that is, T**= T so that becomes T*T- TT* again.
     
  4. Nov 22, 2009 #3
    Ahh thank you very much, I got confused and thought (AB)*=A*B*.
     
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