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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.