Suppose that $H$ is some (complex) Hilbert space and that $\{T_\alpha: \alpha \in I\}$ is some collection of bounded self-adjoint operators on $H$. A version of the spectral theorem states that for each $\alpha$, there exists a measure space $(X,M,\mu)$, a bounded real-valued function $\phi$ and a unitary operator $U: H \to L^2(\mu)$ such that
$T_\alpha = U^*M_\phi U$
where $M_\phi$ denotes the multiplication operator induced by $\phi$. I was wondering under which conditions on $\{T_\alpha\}$ we can find a measure space $(X,M,\mu)$ such that all of the $T_\alpha$ are unitarily equivalent to a multiplication on $L^2(\mu)$? Even better, under what conditions can we choose the same $U$ to work for all $T_\alpha$?
For example (although these are not bounded) all constant coefficient partial differential operators on $L^2(\mathbb{R}^n)$ are Fourier multipliers. What makes them special?