Banach algebras generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols
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ORIGINAL ARTICLE
Banach algebras generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols Miguel Angel Rodriguez Rodriguez1 Received: 8 December 2019 / Accepted: 13 June 2020 Ó Sociedad Matemática Mexicana 2020
Abstract Let D3 be the three-dimensional Siegel domain and A2k ðD3 Þ the weight-ed Bergman space with weight parameter k [ 1. In the present paper, we analyse the commutative (not C ) Banach algebra T ðkÞ generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols acting on A2k ðD3 Þ. We remark that T ðkÞ is not semi-simple, describe its maximal ideal space and the Gelfand map, and show that this algebra is inverse-closed. Keywords Toeplitz operator Weighted Bergman space Commutative Banach algebra Gelfand theory Parabolic quasi-radial quasi-homogeneous
Mathematics Subject Classification Primary 47B35 Secondary 47L80 32A36
1 Introduction One of the most important and interesting questions in the theory of Toeplitz operators is the commutativity of the algebras that they generate. In this regard, we highlight the study of the commutative C -algebras generated by Toeplitz operators acting on the weighted Bergman spaces on the unit ball Bn in [7]. Its main result classifies such algebras as follows: given any maximal commutative subgroup of biholomorphisms of the unit ball, the C -algebra generated by Toeplitz operators, whose symbols are invariant under the action of this group, is commutative on each (commonly considered) weighted Bergman space on Bn . Afterwards, it was quite unexpectedly observed in [10] that there exist many other Banach (not C ) algebras generated by Toeplitz operators that are & Miguel Angel Rodriguez Rodriguez [email protected] 1
Department of Mathematics, CINVESTAV, Mexico City 07360, Mexico
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commutative on each weighted Bergman space. These algebras are induced by classes of symbols subordinated to one of the model classes of the maximal commutative subgroups of biholomorphisms of the unit ball. The detailed structural analysis of such commutative algebras became then an important task as it provides an explicit information on many essential properties of Toeplitz operators such as compactness, boundedness, invariant subspaces, spectral properties, etc. Up to this point, the commutative Banach algebra subordinated to the quasielliptic group has been the best understood one thanks to the work of Bauer and Vasilevski [1–3]. This paper is a further step towards the understanding of these algebras. We approach the problem of describing the commutative Banach algebra subordinated to the quasi-parabolic group for the lowest non-trivial dimension n ¼ 3. We show that this algebra is not semi-simple, describe its maximal ideal space, and show that it is inverse-closed. Consider the Siegel domain D3 . A parabolic (2)-quasi-radial quasi-homogeneous function is a function u 2 L1 ðD3 Þ of the form: p
uðz0 ; z3 Þ ¼ aðr; Imz3 Þfp1 f2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where a ¼ aðr; Imz3 Þ 2 L1 ðD3
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