Toeplitz operators with quasi-radial quasi-homogeneous symbols and bundles of Lagrangian frames

Abstract

We prove that the quasi-homogenous symbols on the projective space ${\mathbb{P}}^n(\mathbb{C})$ yield commutative algebras of Toeplitz operators on all weighted Bergman spaces, thus extending to this compact case known results for the unit ball ${\mathbb{B}}^n$. These algebras are Banach but not $C^{\ast }$. We prove the existence of a strong link between such symbols and algebras with the geometry of ${\mathbb{P}}^n(\mathbb{C})$. The latter is also proved for the corresponding symbols and algebras on ${\mathbb{B}}^n$.

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