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.

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Acknowledgements

I would like to thank Professor Nikolai Vasilevski for his feedback and enlightening comments.

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Correspondence to Miguel Angel Rodriguez Rodriguez.

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Rodriguez, M.A.R. Banach algebras generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols. Bol. Soc. Mat. Mex. 26, 1243–1271 (2020). https://doi.org/10.1007/s40590-020-00299-8

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Keywords

  • Toeplitz operator
  • Weighted Bergman space
  • Commutative Banach algebra
  • Gelfand theory
  • Parabolic quasi-radial quasi-homogeneous

Mathematics Subject Classification

  • Primary 47B35
  • Secondary 47L80
  • 32A36