Abstract
It is known that radial Toeplitz operators acting on a weighted Bergman space of the analytic functions on the unit ball generate a commutative C*-algebra. This algebra has been explicitly described via its identification with the C*-algebra \({{\rm VSO}(\mathbb{N})}\) of bounded very slowly oscillating sequences (these sequences was used by R. Schmidt and other authors in Tauberian theory). On the other hand, it was recently proved that the C*-algebra generated by Toeplitz operators with bounded measurable vertical symbols is unitarily isomorphic to the C*-algebra \({{\rm VSO}(\mathbb{R}_+)}\) of “very slowly oscillating functions”, i.e. the bounded functions that are uniformly continuous with respect to the logarithmic distance \({\rho(x,y)=|\ln(x)-\ln(y)|}\). In this note we show that the results for the radial case can be easily deduced from the results for the vertical one.
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Bauer W., Herrera Yañez C., Vasilevski N.: \({(m,\lambda}\))-Berezin transform and approximation of operators on weighted Bergman spaces over the unit ball. Oper. Theory Adv. Appl. 240, 45–68 (2014)
Bauer W., Herrera Yañez C., Vasilevski N.: Eigenvalue characterization of radial operators on weighted Bergman spaces over the unit ball. Integr. Equ. Oper. Theory 78, 271–300 (2014)
Gradshteyn I.S., Ryzhik I.M.: Tables of Integrals, Series, and Products, 7th edn. Academic Press, New York (1980)
Grudsky S., Karapetyants A., Vasilevski N.: Dynamics of properties of Toeplitz operators on the upper half-plane: parabolic case. J. Oper. Theory 52, 185–204 (2004)
Grudsky S., Quiroga-Barranco R., Vasilevski N.: Commutative C*-algebras of Toeplitz operators and quantization on the unit disk. J. Funct. Anal. 234, 1–44 (2006)
Grudsky S., Vasilevski N.: Bergman-Toeplitz operators: radial component influence. Integr. Equ. Oper. Theory 40, 16–33 (2001)
Grudsky S., Maximenko E., Vasilevski N.: Radial Toeplitz operators on the unit ball and slowly oscillating sequences. Commun. Math. Anal. 14, 77–94 (2013)
Herrera Yañez C., Maximenko E., Vasilevski N.: Vertical Toeplitz operators on the upper half-plane and very slowly oscillating functions. Integr. Equ. Oper. Theory 77, 149–166 (2013)
Herrera Yañez C., Hutník O., Maximenko E.: Vertical symbols, Toeplitz operators on weighted Bergman spaces over the upper half-plane and very slowly oscillating functions. Comptes Rendus Mathematique 352, 129–132 (2014)
Korenblum B., Zhu K.: An application of Tauberian theorems to Toeplitz operators. J. Oper. Theory 33, 353–361 (1995)
Landau E.: Über die Bedeutung einiger neuen Grenzwertsätze der Herren Hardy und Axer. Prace Matematyczno-Fizyczne 21, 97–177 (1910)
Nam K., Zheng D., Zhong C.: m-Berezin transform and compact operators. Rev. Mat. Iberoamericana 22, 867–892 (2006)
Schmidt R.: Über divergente Folgen and lineare Mittelbildungen. Mathematische Zeitschrift 22, 89–152 (1924)
Suárez D.: Approximation and the n-Berezin transform of operators on the Bergman space. J. Reine Angew. Math. 581, 175–192 (2005)
Suárez D.: The eigenvalues of limits of radial Toeplitz operators. Bull. London Math. Soc. 40, 631–641 (2008)
Vasilevski N.: On Bergman-Toeplitz operators with commutative symbol algebras. Integr. Equ. Oper. Theory 34, 107–126 (1999)
Vasilevski N.: Commutative Algebras of Toeplitz Operators on the Bergman Space. Birkhäuser Verlag, Basel (2008)
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This work was partially supported by CONACYT Project 180049 and by IPN-SIP Project 2014-0639, México.
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Herrera Yañez, C., Vasilevski, N. & Maximenko, E.A. Radial Toeplitz Operators Revisited: Discretization of the Vertical Case. Integr. Equ. Oper. Theory 83, 49–60 (2015). https://doi.org/10.1007/s00020-014-2213-2
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DOI: https://doi.org/10.1007/s00020-014-2213-2