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Let $U$ be an operator defined on $l^{2}(\mathbb{Z})$ by $U(e_{n})=e_{n-1}$, where $e_{n}$ is an orthonormal basis of $l^{2}(\mathbb{Z})$. $U$ is a left shift operator. Since $U$ is unitary operator so spectrum is on $S^{1}$. What is spectral measure of $U$? What is its spectral decomposition with respect to multiplicity? Further, suppose T is a self-adjoint operator in B(H) with σ(T) is spectrum of T. μ is a spectral measure. For the operators having general continuous spectrum how to calculate the multiplicity function?

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Identify $l^2(\mathbb{Z})$ with the space of square summable Fourier series $f(z)=\sum_{n\in \mathbb{Z}} a_n z^n$, $\sum |a_n|^2<\infty$, on the unit circle $\mathbb{T}=\{z:|z|=1\}$. It is the space $L^2(\mathbb{T},\lambda)$ where $\lambda$ is Lebesgue measure on the circle and the operator $U$ maps the function $f(z)$ to $z^{-1}f(z)$. It is the spectral decomposition.

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  • $\begingroup$ I don't know every time I am asking a question, people giving negative votes. I wanted to learn from experts over here. What is the problem with Overflowadmin $\endgroup$ – mathlover Feb 16 at 10:55
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    $\begingroup$ Probably people do not consider them being of research level? Try to use math.stackexchange $\endgroup$ – Fedor Petrov Feb 16 at 11:41
  • $\begingroup$ Yaa, sorry, But I already posted this in math stack exchange, after two days I was not getting an answer so I put it here. Please unblock me for giving questions $\endgroup$ – mathlover Feb 16 at 12:07
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    $\begingroup$ You could mention this in the question $\endgroup$ – Fedor Petrov Feb 16 at 13:24

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