Banach algebras generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols

Abstract

Let $$D_3$$ be the three-dimensional Siegel domain and $${\mathcal {A}}_\lambda ^2(D_3)$$ the weight-ed Bergman space with weight parameter $$\lambda >-1$$ . In the present paper, we analyse the commutative (not $$C^*$$ ) Banach algebra $${\mathcal {T}}(\lambda )$$ generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols acting on $${\mathcal {A}}_\lambda ^2(D_3)$$ . We remark that $${\mathcal {T}}(\lambda )$$ is not semi-simple, describe its maximal ideal space and the Gelfand map, and show that this algebra is inverse-closed.