At the moment I am trying to understand the following formulation of the spectral theorem better:
For a bounded, normal operator $T$ on a Hilbert space $H$, the pair $(H,T)$ is unitarily equivalent to $L^2((\Omega, \mu),M)$, where $\mu$ is a positive Baire measure on a locally compact space $\Omega$ and $M$ is the multiplication operator associated with a $\sigma(T)$-valued continuous function on $\Omega$.
I want to apply this theorem to an explicit finite dimensional example to understand it better. Now let T be a $2\times2$-matrix \begin{align} T= \left( \begin{array}{rr} 0 & 2 \\ 2 & 0 \\ \end{array} \right)\, . \end{align} $T$ is bounded and normal and the Hilbert space would be $\mathbb{R}^2$. According to the theorem this pair of $T$ and $\mathbb{R}^2$ is now unitarily equivalent to $L^2((\Omega, \mu),M)$. But I do not really know how to find out what $\mu$, $\Omega$ and $M$ are. How can I find out the right measure and the right space $\Omega$? Does the multiplication $M$ has something to do with the diagonal matrix? I would be really geratful for any help. Thank you very much!