# On spectral multiplicity of left shift operators

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?

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.