# Orthogonal decomposition of $L^2(SM)$

I have been stuck on the following problem for a long time but I could not get the answer. Would you please help me? I was reading one paper [Paternian Salo Uhlmann: Tensor tomography on the surface] in that authors directly write that The space $$L^{2}(S M)$$ decomposes orthogonally as a direct sum $$L^{2}(S M)=\bigoplus_{k \in \mathbb{Z}} H_{k}$$ where $$H_{k}$$ is the eigenspace of $$-i V$$ corresponding to the eigenvalue $$k$$. A function $$u \in L^{2}(S M)$$ has a Fourier series expansion $$u=\sum_{k=-\infty}^{\infty} u_{k}$$ I wanted to justify that. Would you please help me? What I know is the following:

Theory: The standard Hilbert space theory that the complex Hilbert space $$H$$ is said to be a direct-sum of the subspaces $$H_{m}, m \in \mathbb{Z}^{+}$$and we write $$H=\oplus_{m=0}^{\infty} H_{m}$$, if

1.The subspaces $$H_{m}$$ are closed for every $$m$$.

2.The subspaces $$H_{m}$$ are pairwise orthogonal, i.e., if $$h \in H_{m}$$ and $$g \in H_{m^{\prime}}$$, then $$\langle f, g\rangle=0$$ whenever $$m \neq m^{\prime}$$

3.Every element $$f \in H$$ has a representation of the form $$f=h_{0}+h_{1}+\cdots+h_{m}+\cdots$$ where $$h_{m} \in H_{m}\left(m \in \mathbb{Z}^{+}\right)$$and the sum converges in the norm of $$H$$.

The unit sphere bundle is defined as $$S M:=\left\{(x, v) ; x \in M, v \in T_{x} M,|v|_{g}=1\right\}$$.

The vertical vector field $$V$$ which differentiate with respect to direction variable $$v$$ i.e. $$V u(x, v)=\left.\frac{d}{d s} u\left(\rho_{s}(x, v)\right)\right|_{s=0}$$ where $$\rho_{s}(x, v)=\left(x, e^{i s} v\right)$$.

For any $$k \in \mathbb{Z}$$ define $$H_{k}=\left\{u \in L^{2}(S M):-i V u=k u\right\}$$

Any help or hint or reference will be highly appreciated. Thank you so much.
