Algebra generated by Toeplitz operators with \({\mathbb {T}}\)-invariant symbols


We study the structure of the \(C^*\)-algebras generated by Toeplitz operators acting on the weighted Bergman space \({\mathcal {A}}^2_{\lambda }({\mathbb {B}}^2)\) on the two-dimensional unit ball, whose symbols are invariant under the action of the group \({\mathbb {T}}\). We consider three principally different basic cases of its action \(t:\,(z_1,z_2) \mapsto (tz_1,t^{k_2}z_2)\), with \(k_2=1,0,-1\). The properties of the corresponding Toeplitz operators as well as the structure of the \(C^*\)-algebra generated by them turn out to be drastically different for these three cases.

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  1. 1.

    Alzer, H.: Inequalities for the Beta function of \(n\) variables. ANZIAM J. 44, 609–623 (2003)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Bauer, W., Hagger, R., Vasilevski, N.: Algebras of Toeplitz operators on the n-dimensional unit ball. Complex Anal. Oper. Theory 13(2), 493–524 (2019)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Bauer, W., Vasilevski, N.: On algebras generated by Toeplitz operators and their representations. J. Funct. Anal. 272, 705–737 (2017)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Bunce, J.W., Deddens, J.A.: \(C^{\ast } \)-algebras generated by weighted shifts. Indiana Univ. Math. J. 23, 257–271 (1973/1974)

  5. 5.

    Coburn, L.: The \(C^{\ast } \)-algebra generated by an isometry. Bull. Am. Math. Soc. 73, 722–726 (1967)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products. Academic, New York (1980)

    MATH  Google Scholar 

  7. 7.

    Grudsky, S.M., Maximenko, E.A., Vasilevski, N.L.: Radial Toeplitz operators on the unit ball and slowly oscillating sequences. Commun. Math. Anal. 14(2), 77–94 (2013)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Herrera Yañez, C., Maximenko, E.A., Vasilevski, N.: Radial Toeplitz operators revisited: discretization of the vertical case. Integral Equ. Oper. Theory 83(1), 49–60 (2015)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Quiroga-Barranco, R., Vasilevski, N.: Commutative \(C^*\)-algebras of Toeplitz operators on the unit ball, I. Bargmann-type transforms and spectral representations of Toeplitz operators. Integral Equ. Oper. Theory 59(3), 379–419 (2007)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Schmidt, R.: Über divergente Folgen and lineare Mittelbildungeno. Math. Z. 22, 89–152 (1925)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Suárez, D.: The eigenvalues of limits of radial Toeplitz operators. Bull. Lond. Math. Soc. 40, 631–641 (2008)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Sudo, T.: The spectrum theory for weighted shift operators and their \(C^*\)-algebras. Bull. Fac. Sci. Univ. Ryukyus 106, 1–31 (2018)

    MathSciNet  Google Scholar 

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This work was partially supported by CONACYT Project 238630, México.

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Correspondence to Nikolai Vasilevski.

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Vasilevski, N. Algebra generated by Toeplitz operators with \({\mathbb {T}}\)-invariant symbols. Bol. Soc. Mat. Mex. 26, 1217–1242 (2020).

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  • Toeplitz operators
  • \({\mathbb {T}}\)-invariant symbol
  • Bergman space

Mathematics Subject Classification

  • Primary 47B35
  • Secondary 32A36