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Adjoint of linear Operator and T-invariant subspace

  1. Oct 30, 2011 #1
    1. The problem statement, all variables and given/known data

    Let T be a linear operator on an inner product space V and W be a T-invariant subspace of V. If W is both T and T* invariant, prove that (T[itex]_{W}[/itex])* = (T*)[itex]_{W}[/itex]. Note that T[itex]_{W}[/itex] denotes the restriction of T to W

    2. Relevant equations

    [itex]\forall[/itex]x[itex]\in[/itex]W, T[itex]_{W}[/itex](x) = T(x)
    T(W)[itex]\subseteq[/itex]W
    T*(W)[itex]\subseteq[/itex]W

    3. The attempt at a solution

    I attempt to show that the image of any x [itex]\in[/itex]W under (T[itex]_{W}[/itex])* and (T*)[itex]_{W}[/itex] are the same.
    <T[itex]_{W}[/itex]*(x), y> = < x, T[itex]_{W}[/itex](y)> = < x, T(y) > = < T*(x), y> = <(T*)[itex]_{W}[/itex](x), y >

    Since this is true for all y, therefore, (T[itex]_{W}[/itex])* = (T*)[itex]_{W}[/itex].

    Is the above proof correct? I feel that there is something wrong in one of the steps.
    Thank You.
     
  2. jcsd
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