1. 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

2. Originally Posted by Mauritzvdworm
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 $\displaystyle x_n = \left[\begin{smallmatrix}0 \\e_n \end{smallmatrix}\right]$.