Toeplitz Operators with $$\mathbb {T}_q^m$$ -Invariant Symbols on Some Weakly Pseudoconvex Domains

Abstract

In this paper, we study the Banach algebra \(\mathcal {T}(\mathbb {T}_m^q)\) which is generated by Toeplitz operators whose symbols are invariant under the action of the \(\mathbb {T}_m^q\) subgroup of the maximal torus \(\mathbb {T}^n\), which are acting on the Bergman space on weakly pseudo-convex domains \( \Omega ^n_p\). Moreover, we proved that the commutator of the C∗-algebra \(\mathcal {T}(\mathcal {R}_k(\Omega ^n_p))\) is equal to the Toeplitz algebra \(\mathcal {T}(\mathbb {T}_m^q)\), where \(\mathcal {T}(\mathcal {R}_k(\Omega ^n_p))\) is the C∗-algebra generated by Toeplitz operators where the symbols are k-quasi-radial. Finally, using this relationship we found some commutative Banach algebras generated by Toeplitz operators which generalize the Banach algebra generated by Toeplitz operators with quasi-homogeneous quasi-radial symbols.