Exercise 8, § 1

The group $\mathbf{R}/\mathbf{Z}$ is provided with its normalized Haar measure $\mu$. a) Let $h\in \mathscr{L}^2(\mathbf{R}/\mathbf{Z})$. For $f\in \mathscr{L}^2(\mathbf{R}/\mathbf{Z})$, define a function $C_h(f)$ on $\mathbf{R}/\mathbf{Z}$ by the formula

The mapping $f\mapsto C_h(f)$ is a continuous endomorphism of $\mathscr{L}^2(\mathbf{R}/\mathbf{Z})$, and defines by passing to quotients a compact normal endomorphism of $L^2(\mathbf{R}/\mathbf{Z})$.

b) Determine the spectrum of $C_h$ and the spectral multiplicity of the elements of this spectrum. (Develop $h$ in a Fourier series.)

c) There exists $h\in \mathscr{C}(\mathbf{R}/\mathbf{Z})$ such that $C_h$ is not of finite trace.