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Integral Equations and Operator Theory

, Volume 34, Issue 1, pp 107–126 | Cite as

On Bergman-Toeplitz operators with commutative symbol algebras

  • N. L. Vasilevski
Article

Abstract

Let\(\mathbb{D}\) be the unit disk inℂ,\(\mathcal{A}^2 (\mathbb{D})\) be the Bergman space, consisting of all analytic functions from\(L_2 (\mathbb{D})\), and\(B_\mathbb{D} \) be the Bergman projection of\(L_2 (\mathbb{D})\) onto\(\mathcal{A}^2 (\mathbb{D})\). We constructC*-algebras\(\mathcal{A} \subset L_\infty (\mathbb{D})\), for functions of which the commutator of Toeplitz operators [T a ,T b ]=T a T b −T b T a is compact, and, at the same time, the semi-commutator [T a ,T b )=T a T b −T ab is not compact.

It is proved, that for each finite set ∧=〈n0,n1, ...,n m , where 1=n0<n1<...<n m ≤∞, andn k ∈ℕ∪ {∞}, there are algebras\(\mathcal{A}_\Lambda \) of the above type, such that the symbol algebras Sym\(\mathcal{T}(\mathcal{A}_\Lambda )\) of Toeplitz operator algebras\(\mathcal{T}(\mathcal{A}_\Lambda )\) arecommutative, while the symbol algebras Sym\(\mathcal{R}(\mathcal{A}_\Lambda ,B_\mathbb{D} )\) of the algebras\(\mathcal{R}(\mathcal{A}_\Lambda ,B_\mathbb{D} )\), generated by multiplication operators\(a \in \mathcal{A}_\Lambda \) and\(B_\mathbb{D} \), haveirreducible representations exactly of dimensions n0,n1,..., n m .

AMS Classification

47B35 47D25 

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Copyright information

© Birkhäuser Verlag 1999

Authors and Affiliations

  • N. L. Vasilevski
    • 1
  1. 1.Departamento de MatemáticasCINVESTAV del I.P.N.MexicoMexico

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