Let $A$ be a $C^*$-algebra, $E$ a Hilbert $A$-module and $T:E\rightarrow E$ a bounded self-adjoint regular operator. Is there something like a spectral theorem for Hilbert modules that gives a decomposition of $E$ in relation to the spectrum of $T$? For example, is there an analogue of the spectral theorem for compact self-adjoint operators?
Edit: clarified question with Martin's comments below.