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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2504.11600 |
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| _version_ | 1866909845617115136 |
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| author | Andruchow, Esteban |
| author_facet | Andruchow, Esteban |
| contents | Let $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$ and $\mathbb{T}=\{z\in\mathbb{C}: |z|=1\}$. For $a\in\mathbb{D}$, consider $φ_a(z)=\frac{a-z}{1-\bar{a}z}$ and $C_a$ the composition operator in $L^2(\mathbb{T})$ induced by $φ_a$: $$ C_a f=f\circφ_a. $$ 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 $a\ne b\in\mathbb{D}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11600 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reflections in $L^2(\mathbb{T})$ Andruchow, Esteban Functional Analysis 47A05, 47B33, 47B91 Let $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$ and $\mathbb{T}=\{z\in\mathbb{C}: |z|=1\}$. For $a\in\mathbb{D}$, consider $φ_a(z)=\frac{a-z}{1-\bar{a}z}$ and $C_a$ the composition operator in $L^2(\mathbb{T})$ induced by $φ_a$: $$ C_a f=f\circφ_a. $$ 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 $a\ne b\in\mathbb{D}$. |
| title | Reflections in $L^2(\mathbb{T})$ |
| topic | Functional Analysis 47A05, 47B33, 47B91 |
| url | https://arxiv.org/abs/2504.11600 |