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On commutative C-algebras generated by Toeplitz operators with \( \mathbb{T}^{m}\)-invariant symbols

  • Nikolai Vasilevski
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 271)

Abstract

It is known that Toeplitz operators, whose symbols are invariant under the action a maximal Abelian subgroups of biholomorphisms of the unit ball \( \mathbb{B}^{n}\), generate the C-algebra being commutative in each standardly weighted Bergman space. In case of the unit disk (n = 1) this condition on generating symbols is also necessary in order that the corresponding C-algebra be commutative. In this paper, for n > 1, we describe a wide class of symbols that are not invariant under the action of any maximal Abelian subgroup of biholomorphisms of the unit ball, and which, nevertheless, generate via corresponding Toeplitz operators C-algebras being commutative in each standardly weighted Bergman space. These classes of symbols are certain proper subsets of functions that are invariant under the action of the group \( \mathbb{T}^{m}\), with mn, being a subgroup of the maximal Abelian subgroup \( \mathbb{T}^{n}\) of biholomorphisms of \( \mathbb{B}^{n}\).

Keywords

Toeplitz operators Bergman space commutative C-algebras 

Mathematics Subject Classification (2010)

Primary 47B35 Secondary 47L80 32A36 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Departamento de Matem´aticas, CINVESTAVApartado Postal 14-740MéxicoMéxico

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