2020/09/25 by Raúl Quiroga-Barranco, Quiroga-Barranco, Raul, Armando Sánchez-Nungaray +1
Mathematics · #32A36 #47B35 #53D20 #Advanced Topics in Algebra #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2009.12448
openalex publication_date 2020/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that to every connected Abelian subgroup H of the biholomorphisms\nof the unit ball mathbbBn we can associate a set of bounded symbols whose\ncorresponding Toeplitz operators generate a commutative C^*-algebra on every\nweighted Bergman space. These symbols are of the form a(z) = f(\μH(z)),\nwhere \μH is the moment map for the action of H on mathbbBn. We\nshow that, for this construction, if H is a maximal Abelian subgroup, then\nthe symbols introduced are precisely the H-invariant symbols. We provide the\nexplicit computation of moment maps to obtain special sets of symbols described\nin terms of coordinates. In particular, it is proved that our symbol sets have\nas particular cases all symbol sets from the current literature that yield\nToeplitz operators generating commutative C^*-algebras on all weighted\nBergman spaces on the unit ball mathbbBn. Furthermore, we exhibit\nexamples that show that some of the symbol sets introduced in this work have\nnot been considered before. Finally, several explicit formulas for the\ncorresponding spectra of the Toeplitz operators are presented. These include\nspectral integral expressions that simplify the known formulas for maximal\nAbelian subgroups for the unit ball.\n