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?



On spectral multiplicity of left shift operators
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.


