1
$\begingroup$

We define $H=l^2(\mathbb{Z})$, $S\in L(H)$ to be the left (bilateral) shift and we look at $T=S+S^*$ ($S^*$ is actually the right shift). We need to prove that the spectrum of $T$ is $[-2, 2]$ and more importantly to find a unitary operator and 2 measures $\mu_1$ and $\mu_2$ in order to represent $T$ as multiplication by $x$ on $L^2(\mathbb{R}, \mu_1)\bigoplus L^2(\mathbb{R},\mu_2)$. Help would be greatly appreciated.

1 Answers 1