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.