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.