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Math Help - Adjoint of linear Operator

  1. #1
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    Adjoint of linear Operator

    Let V be an inner product space and y,z be in V. Let T be a linear operator on V defined by T(x) = <x,y>z for all x in V. Find the explicit expression for T*.

    I know that <T(x),w> = <x,T*(w)>.
    Hence, <<x,y>z,w> = <x,T*(w)>. But <x,y> is just a scalar.
    Then, <x,y><z,w> = <x,T*(w)>.

    But how do i proceed on from here to find T*?
    Thank you.
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  2. #2
    MHF Contributor

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    Re: Adjoint of linear Operator

    Is your vector space over the real or complex numbers? That's important because for a space over the real numbers, <x, y> = <y x> while for a space over the complex numbers, <x,y>= <y, x>* where * is the complex conjugate.

    If over the real numbers, then <x, y><z, w>= <x, y<z, w>> and, since that is to be equal to < x, T*(w)>, T*(w)= <z, w>y.
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