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I am trying to understand the proof of the spectral theorem for bounded, selfadjoint operators on a Hilbertspace $H$ in the book 'Functional Analysis' from Dirk Werner.

The structure of the proof goes as follows.

First, it is shown that if $T \in L^2(H)$ is selfadjoint and it exists a $x\in H$ with $$H=\overline{span\{T^nx\mid n\geq 0 \}}$$ then there exists a measure $\mu$ and a unitary operator $U : H\to L^2(\mu)$ with $$ (U T U^{-1} \phi)(t) = t \cdot \phi(t).$$

Then, it is shown that if $H$ is separable, one can find pairwise orthogonal subspaces $H_i$ with $$H_i=\overline{span\{T^nx\mid n\geq 0 \}}$$ for some $x\in H_i$ and $T(H_i) \subseteq H_i$ such that $$H= \oplus H_i$$ holds.

Now to prove the gerenal case, the author applies the lemma from the first step on each $T_i : H_i \to H_i$ and states:

There exists a unitary map $U_i: H_i \to L^2(\mu_i)$ and a bounded measurable function $$f_i : \sigma(T_i) \to \mathbb{R}$$ with $$ U_i T_i U_i^{-1} (\phi_i) = f_i \, \phi_i.$$ $$[..]$$ $$ \text{define } f(t) = f_i(t) \text{ for } t \in \sigma(T_i) $$ $$[...]$$ $$UTU^{-1} (\phi) = f \phi$$

My questions are:

  1. Why does he not say $f_i(t)=t$ and writes $f(t) = t$?
  2. If there is really a reason to write $f_i$ instead of the identity, how do I see that $f$ is well defined? Would it not be possible that $t\in \sigma(T_i)$ and $t \in \sigma(T_j)$ for some $i\neq j$?

Thank you for any advice.

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