Reflections in $L^2(\mathbb{T})$
Esteban Andruchow
TL;DR
This paper analyzes composition operators C_a on L^2(T) induced by disk automorphisms and their associated symmetries R_a and W_a. It provides explicit descriptions of the eigenspaces N(T ± I) for T ∈ {C_a, R_a, W_a}, using fixed points ω_a and Ω_a of the automorphisms and the Szego kernel, and expresses these eigenspaces in terms of even/odd subspaces and Hardy space components. A core result is that for a ≠ b, N(C_a − I) ∩ N(C_b − I) = ⟨1⟩ and N(C_a + I) ∩ N(C_b + I) = {0}, and the two eigenspaces N(C_a − I) and N(C_a + I) are in generic position with no nontrivial intersections. By leveraging R_a and W_a, the authors transfer intersections and deduce several explicit relations, which illuminate the Grassmannian geometry of Gr(H) and yield criteria for joining subspaces by unique normalized geodesics. Altogether, the work provides a detailed, parameter dependent Grassmann-geometry framework for symmetry-induced decompositions of L^2(T) under disk automorphisms.
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 $a\ne b\in\mathbb{D}$.