I am looking for an operator $T$ between the spaces of smooth functions over some interval $[a,b]$ such that for any function $f$,
$$T(T(f)) = f$$
and $T$ is self-adjoint: $$\langle f | T g \rangle = \langle T f | g \rangle$$
where the dot product is specified as $\langle f | g \rangle = \int_a^b f^*(x) g(x) dx$.
I am aware of simple operators such as $f(x) \to -f(x)$, $f(x) \to f(-x)$, $f(x) \to -f(-x)$.
Background: Chiral symmetry, a type of symmetry in quantum mechanics, arises from invariance under a transformation that is unitary and involutary. These transformations are generally studied in the context of finite dimensional vector spaces, where common examples are the Pauli matrices. I was wondering if it was possible to generalize this to function spaces.
The only linear possibilities are going to be unitary "half-rotations", and reflections across an eigenspace. Consider the Fractional Fourier Transform for $\alpha = \pi$ on the rotation front:
$$\begin{align}[F_\pi f](\xi)&=\int_{-\infty}^\infty f(x)\delta(x\pm\xi)dx\\ &=f(\pm\xi) \end{align} $$ In terms of reflection:
Define "hyperplane" $W=(g(x))^\perp$ of the function space, $$\begin{align} R_{g}f&=f-2\frac{\left\langle f,g \right\rangle}{\left\langle g,g \right\rangle}g\\ &=f-2\frac{\int_a^b f^*(x) g(x) dx}{\int_a^b g^*(x) g(x) dx}g \end{align} $$