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Math Help - Finding the adjoint

  1. #1
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    Finding the adjoint

    Given the operator  T: L^2([0,1]) \rightarrow L^2([0,1]) defined by
    (Tf)(x) = \int^x_0 f(t) \ dt
    How do I find adjoint of T? I'm totally lost as to how to go about doing this. Thanks in advance!
    Last edited by anegligibleperson; August 14th 2012 at 11:58 AM.
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  2. #2
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    Re: Finding the adjoint

    In an inner product space, the "adjoint" of a linear transformation, A, is the linear transformation A* such that <Au, v>= <u, A*v>. Here the inner product is <u(x), v(x)>= \int_0^1 u(x)v(x)dx. So you are looking for A* such that <Au, v>= \int_0^1 \left(\int_0^x u(t)dt\right)v(x)dx= \int_0^1 u(x)A*(v(x))dx. Do the integrals on the left and find the function A* that makes the right side equal to that. I would suggest using "integration by parts".
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  3. #3
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    Re: Finding the adjoint

    @HallsofIvy, could you be more explicit? I get  \int_0^xu(t)dtV(x)|_0^1 - \int_0^1u(x)V(x)dx then I don't know how to proceed.
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  4. #4
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    Re: Finding the adjoint

    You actually obtain

    \int_0^1 U(x)v(x)dx = U(x)V(x)|_0^1-\int_0^1u(x)V(x)dx

    That is

    \langle u, T^*v\rangle = \langle Tu,v\rangle = U(x)V(x)|_0^1 - \langle u,Tv\rangle

    This suggest that you need to write the product U(x)V(x) in a form with u in it. That is, you need to make it look more like \langle u,Tv\rangle and somehow combine the two. I hope this helps.
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