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Math Help - show operator not compact

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

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

    \[ \left( \begin{array}{cc}<br />
0 & V \\<br />
V^* & 0 \end{array} \right)\] is unitary

    Show that
    \lambda \mathbb{I}-\left( \begin{array}{cc}<br />
0 & V \\<br />
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 H=l^2\oplus l^2 and choose a unitary operator V on l^2

    \[ \left( \begin{array}{cc}<br />
0 & V \\<br />
V^* & 0 \end{array} \right)\] is unitary

    Show that
    \lambda \mathbb{I}-\left( \begin{array}{cc}<br />
0 & V \\<br />
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 (x_n) in H and see what \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 (e_n) for l^2 and put x_n = \left[\begin{smallmatrix}0 \\e_n \end{smallmatrix}\right].
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