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

Abstract

Let $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$ and $\mathbb{T}=\{z\in\mathbb{C}: |z|=1\}$. For $a\in\mathbb{D}$, consider $\varphi_a(z)=\frac{a-z}{1-\bar{a}z}$ and $C_a$ the composition operator in $L^2(\mathbb{T})$ induced by $\varphi_a$: $$ C_a f=f\circ\varphi_a. $$ 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 $C_a$: $$ R_a=M_{\frac{|k_a|}{\|k_a\|_2}}C_a \ \hbox{ and } \ W_a=M_{\frac{k_a}{\|k_a\|_2}}C_a, $$ 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 $a\ne b\in\mathbb{D}$.