Symmetries and reflections from composition operators in the disk
Esteban Andruchow, Gustavo Corach, Lázaro Recht
TL;DR
The paper analyzes composition operators $C_a f=f\circ\varphi_a$ on the Hardy space $H^2$ of the unit disk, where $\varphi_a(z)=(a-z)/(1-\bar{a}z)$ and $|a|<1$, showing these operators are reflections with $C_a^2=I$. It characterizes the eigenspaces $N(C_a\pm I)$ via the fixed point $\omega_a$ of $\varphi_a$ as $N(C_a-I)=C_{\omega_a}({\cal E})$ and $N(C_a+I)=C_{\omega_a}({\cal O})$, where ${\cal E},{\cal O}$ are the even/odd subspaces of $H^2$. The symmetry $\rho_a$ (the unitary part of the polar decomposition) is studied in depth, with explicit links to $W_a$, Toeplitz operators, and Rosenblum’s spectral measure, yielding local injectivity of $a\mapsto \rho_a$ and formulas for the range and nullspace symmetries. The paper also investigates the geometry of the eigenspaces on the Grassmann manifold, proving that no geodesic exists between $N(C_a-I)$ and $N(C_a+I)$ for $a\neq 0$, while establishing geodesics between certain mixed pairs (e.g., $\mathcal{E}$ with $N(C_a-I)$) and characterizing intersection properties. Overall, the results illuminate the interplay between disk geometry, operator symmetries, and Grassmannian geometry for these canonical reflections.
Abstract
We study the composition operators $C_a$ acting on the Hardy space $H^2$ of the unit disk, given by $C_af=f\circ\varphi_a$, where $$ \varphi_a(z)=\frac{a-z}{1-\bar{a}z}, $$ for $|a|<1$. These operators are reflections: $C_a^2=1$. We study their eigenspaces $N(C_a\pm 1)$, their relative position (i.e., the intersections between these spaces and their orthogonal complementes for $a\ne b$ in the unit disk) and the symmetries induced by $C_a$ and these eigenspaces.