There is a (weaker) version of spectral theorem saying that any self-adjoint operator in Hilbert space is unitarily isomorphic to multiplication operator in $L_2(X,\mu)$, where $(X,\mu)$ is some measurable space.
How can we find such a representation, for example, for a diagonal operator? I mean, how to choose $X,\mu,$ the function $f$ for multiplication operator and the unitary isomorphism needed for unitary equivalence?
Thank you.
I take it a "diagonal operator" is $T:\ell_2\to\ell_2$ given by $$T(x_1,x_2,\dots)=(m_1x_1,m_2,m_2,\dots)$$for some sequence $m_j$. This means that a diagonal operator is a multiplication operator! Let $X=\Bbb N$, let $\mu$ be counting measure on $\Bbb N$ and define $U:\ell_2\to L^2(X,\mu)$ by $$Ux=x.$$Then $Tx=U^{-1}mUx$.