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In the book Analysis Now by Pedersen, the Spectral Theorem is that, for a normal operator $T$ acting on a Hilbert space $H$, there is an isometric star-isomorphism between $C(\text{sp}(T))$ and the $C^*$-algebra that is generated by $I$ and $T$. This star-isomorphism is called the continous functional calculus for $T$.

I am under the impression that this is the first -- or at least an early -- version of the Spectral Theorem (for the infinite-dimensional setting). First, what does this tell us, that is, why would one care about a functional calculus? Second, how does this relate to the more common multiplication-version of the the Spectral Theorem?

LinearGuy
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  • I'm a bit tired now but the functional calculus allows you to construct elements in your C*-algebra that behave like the corresponding functions do. For example, you can prove that for an element $x$ there is an element $y$ with $x=y^2$. Here we thus have mimiqued the behaviour of the root-function. If you'd like, I can expand on this a bit in an answer. – J. De Ro Apr 01 '20 at 21:44
  • One finds an answer here. Roughly, if one could approximate characteristic functions of measurable subsets of sp$(T)$ by continuous functions, then one would obtain the resolution of the identity/projector valued measure. Still thinking about this data and the "change of basis" version... – Noix07 Aug 11 '22 at 14:53

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The first spectral theorem was von Neumann's representation of a self-adjoint operator $A$ $$ Ax = \int_{-\infty}^{\infty} \lambda dE(\lambda)x $$ where $E(\lambda)$ is a non-decreasing orthogonal projection-valued function of $t$ on $\mathbb{R}$ and $x\in\mathcal{D}(A)$.

Earlier specialized versions were aimed at understanding the integral and discrete eigenfunction expansions associated with Sturm-Liouville ODE problems, and with the partial differential equations that gave rise to the Sturm-Liouville problems through separation of variables.

Disintegrating By Parts
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