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Thread: show operator not compact

  1. #1
    Member Mauritzvdworm's Avatar
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    show operator not compact

    Consider the Hilbert space $\displaystyle H=l^2\oplus l^2$ and choose a unitary operator $\displaystyle V$ on $\displaystyle l^2$

    $\displaystyle \[ \left( \begin{array}{cc}
    0 & V \\
    V^* & 0 \end{array} \right)\]$ is unitary

    Show that
    $\displaystyle \lambda \mathbb{I}-\left( \begin{array}{cc}
    0 & V \\
    V^* & 0 \end{array} \right)$ is never compact
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by Mauritzvdworm View Post
    Consider the Hilbert space $\displaystyle H=l^2\oplus l^2$ and choose a unitary operator $\displaystyle V$ on $\displaystyle l^2$

    $\displaystyle \[ \left( \begin{array}{cc}
    0 & V \\
    V^* & 0 \end{array} \right)\]$ is unitary

    Show that
    $\displaystyle \lambda \mathbb{I}-\left( \begin{array}{cc}
    0 & V \\
    V^* & 0 \end{array} \right)$ is never compact
    One property of a compact operator is that it takes each bounded sequence to a sequence with a convergent subsequence. So take a bounded sequence $\displaystyle (x_n)$ in H and see what $\displaystyle \lambda \mathbb{I}-\left[\begin{smallmatrix}0 & V \\V^* & 0 \end{smallmatrix}\right]$ does to it. For example, you could start with an orthonormal basis $\displaystyle (e_n)$ for $\displaystyle l^2$ and put $\displaystyle x_n = \left[\begin{smallmatrix}0 \\e_n \end{smallmatrix}\right]$.
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