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Math Help - example problem

  1. #1
    needhelp_thanks
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    example problem

    Would someone help me with this problem?

    Give an example of a linear operator T on an inner product space V such that N(T) does not equal N(T*).
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Is N(T) supposed to be the null space of T, or the dimension of the null space?

    If N(T) = null space of T, then you can get an example in two dimensions. Take T=\begin{bmatrix}0&1\\0&0\end{bmatrix}.

    If N(T) means the nullity of T then you have to go to an infinite-dimensional space. The simplest example would be the unilateral shift on \ell_2. This is 1-1, so has zero-dimensional null space. But its adjoint is the backwards shift, which has one-dimensional kernel.
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