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Linear Operator such that it is Zero Operator

  1. Apr 15, 2010 #1
    1. The problem statement, all variables and given/known data
    Let T, linear operator on non fin. dim. Vector Space over field of complex numbers such that there exists T*, adjoint of T.

    If <T(x),x> = 0 for all x in V then T=T_0

    T_0 s.t. T_0(x)=0 for all x in V

    2. The attempt at a solution
    Consider <T(x''),x''> s.t. x''=x' + y

    <T(x''),x''> = <T(x'+y),x'+y> = <T(x')+T(y),x' + y> = <T(x'),x'> + <T(x'),y> + <T(y), x'> + <T(y),y)>

    Then <T(x'),x'> = <T(y),y)> = 0

    Let x' = x + iy

    <T(x+iy),y> + <T(y),x+iy> = <T(x),y> + <iT(y),y> + <T(y),x> + <T(y),iy>

    Then <iT(y),y> + <T(y),iy> = i<T(y),y> + -i<T(y),y> = 0

    Then we have <T(x),y> +<T(y),x> = 0

    I assume what we want to get is something like <a,a> = 0...then this is only true if a=0...i.e. T(x)=0 for all x in V

    Is it the case of defining x'' and x' in the manner I did that we can assume x and y are real now? I.E. <T(y),x> = <x,T(y)> Even with this assumption I don't see the answer.
    I'm not sure how to finish...any hints? Is my procedure up to this point what I should have done?

    Thanks.
     
  2. jcsd
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