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