Reflections in $L^2(\mathbb{T})$

Let $\mathbb{D}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$ and $\mathbb{T}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$. For $a\in \mathbb{D}$, consider ${\phi }_a(z)=\frac{a-z}{1-\bar{a}z}$ and $C_a$ the composition operator in $L^2(\mathbb{T})$ induced by ${\phi }_a$: $$C_{a}f=f\circ {\phi }_a\mathrm{.}$$ Clearly $C_a$ satisties $C_a^2=I$, i.e., is a non-selfadjoint reflection. We also consider the following symmetries (selfadjoint reflections) related to $C_a$: $$R_a=M_{\frac{\mathrm{\mid }{k}_a\mathrm{\mid }}{\mathrm{\parallel }{k}_a{\mathrm{\parallel }}_2}}{C}_a\text{ and }{W}_a=M_{\frac{k_a}{\mathrm{\parallel }{k}_a{\mathrm{\parallel }}_2}}{C}_a,$$ where $k_a(z)=\frac{1}{1-\bar{a}z}$ is the Szego kernel. The symmetry $R_a$ is the unitary part in the polar decomposition of $C_a$. We characterize the eigenspaces $N(T_a\pm I)$ for $T_a=C_a,R_a$ or $W_a$, and study their relative positions when one changes the parameter $a$, e.g., $N(T_a\pm I)\cap N(T_b\pm I)$, $N(T_a\pm I)\cap N(T_b\pm I{)}^{\perp }$, $N(T_a\pm I{)}^{\perp }\cap N(T_b\pm I)$, etc., for $a\mathrm{\ne }b\in \mathbb{D}$.