Reflections in L2(T)

Abstract

Let $D=\{z\in C:|z|<1\}$ and $T=\{z\in C:|z|=1\}$. For $a\in D$, consider ${\phi }_a\left(z\right)=\frac{a-z}{1-\overline{a}z}$ and $C_a$ the composition operator in $L^2\left(T\right)$ induced by ${\phi }_a$:$C_{a}f=f\circ {\phi }_a.$ Clearly $C_a$ satisfies $C_a^2=I$, i.e., is a non-selfadjoint reflection. We also consider the following symmetries (selfadjoint reflections) related to $C_a$:$R_a=M_{\frac{\left|k_a\right|}{{\Vert k_a\Vert }_2}}{C}_a\text{ and }{W}_a=M_{\frac{k_a}{{\Vert k_a\Vert }_2}}{C}_a,$ where $k_a\left(z\right)=\frac{1}{1-\overline{a}z}$ 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 $a\ne b\in D$.