# Thread: show operator not compact

1. ## show operator not compact

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

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

Show that
$\lambda \mathbb{I}-\left( \begin{array}{cc}
0 & V \\
V^* & 0 \end{array} \right)$
is never compact

2. Originally Posted by Mauritzvdworm
Consider the Hilbert space $H=l^2\oplus l^2$ and choose a unitary operator $V$ on $l^2$

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

Show that
$\lambda \mathbb{I}-\left( \begin{array}{cc}
0 & V \\
V^* & 0 \end{array} \right)$
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 $(x_n)$ 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 $x_n = \left[\begin{smallmatrix}0 \\e_n \end{smallmatrix}\right]$.