Toeplitz operators on quotient domains

Date/heure
21 septembre 2023
14:15 - 15:15

Lieu
Salle de conférences Nancy

Oratrice ou orateur
E. K. Narayanan (Indian Institute of Science)

Catégorie d'évènement
Séminaire Théorie de Lie, Géométrie et Analyse

Let $G$ be a finite pseudo-reflection group and $\mathrm{\Omega }$ be a bounded domain in ${\mathbb{C}}^d$ which is $G$-invariant. The quotient domain $\mathrm{\Omega }/G,$ is biholomorphically equivalent to a domain $\mathit{\theta }(\mathrm{\Omega })$ where $\mathit{\theta }:\mathrm{\Omega }\to \mathit{\theta }(\mathrm{\Omega })$ is a basic polynomial map. Prominent example of a quotient domain is the symmetrized polydisc ${\mathbb{G}}_d$ in ${\mathbb{C}}^d.$ In this case, the basic polynomial map is given by $z\to (s_1(z),s_2(z),\cdots s_d(z))$ from ${\mathbb{D}}^d$ (unit polydisc in ${\mathbb{C}}^d$) to ${\mathbb{G}}_d$ where $s_j(z)$ is the $j$-th elementary symmetric polynomial. We study properties of Toeplitz operators on weighted Bergman spaces on $\mathit{\theta }(\mathrm{\Omega })$ by establishing a connection of them with Toeplitz operators on weighted Bergman spaces on $\mathrm{\Omega }.$ Results on zero product problem and commuting pairs of Toeplitz operators will be explained. Representation theory of $G$ and projections to isotypic components play an important role in our results. (Joint work with Gargi Ghosh)