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Algebraic Properties of Toeplitz Operators on the Symmetrized Polydisk

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Abstract

This paper is an effort to continue the legacy of the classically successful theory of Toeplitz operators on the Hardy space over the unit disk to a new domain in \({\mathbb {C}}^d\)—the symmetrized polydisk. We obtain algebraic characterizations of Toeplitz operators, analytic Toeplitz operators, compact perturbation of Toeplitz operators and dual Toeplitz operators. We then revisit the operator theory of this domain considered first in Biswas and Shyam Roy (J Funct Anal 266:6224–6255, 2014), to study the generalized Toeplitz operators and find a commutant lifting type result.

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References

  1. Agler, J., Lykova, Z., Young, N.J.: Geodesics, retracts, and the norm-preserving extension property in the symmetrized bidisc, Mem. Am. Math. Soc. 258 (2019), no. 1242, vii+108 pp. ISBN: 978-1-4704-3549-3

  2. Agler, J., Young, N.J.: The hyperbolic geometry of the symmetrized bidisc. J. Geom. Anal. 14, 375–403 (2004)

    Article  MathSciNet  Google Scholar 

  3. Agler, J., Young, N.J.: A model theory for \(\Gamma \)-contractions. J. Operator Theory 49, 45–60 (2003)

    MathSciNet  MATH  Google Scholar 

  4. Aherna, P., Cuckovic, Z.: A theorem of Brown–Halmos type for Bergman space Toeplitz operators. J. Funct. Anal. 187, 200–210 (2001)

    Article  MathSciNet  Google Scholar 

  5. Alexander, H., Wermer, J.: Several Complex Variables and Banach Algebras. Springer, New York (1998)

    MATH  Google Scholar 

  6. Axler, S., Conway, J.B., McDonald, G.: Toeplitz operators on Bergman spaces. Can. J. Math. 34, 466–483 (1982)

    Article  MathSciNet  Google Scholar 

  7. Bhattacharyya, T., Das, B.K., Sau, H.: Toeplitz operators on the symmetrized bidisc. Int. Math. Res. Not. IMRN (to appear). https://doi.org/10.1093/imrn/rnz333, available at arXiv:1706.03463

  8. Bhattacharyya, T., Sau, H.: Holomorphic functions on the symmetrized bidisk—realization, interpolation and extension. J. Funct. Anal. 274(2), 504–524 (2018)

    Article  MathSciNet  Google Scholar 

  9. Biswas, S., Shyam Roy, S.: Functional models of \(\Gamma _d\)-contractions and characterizations of \(\Gamma _d\)-isometries. J. Funct. Anal. 266, 6224–6255 (2014)

    Article  MathSciNet  Google Scholar 

  10. Böttcher, A., Silbermann, B.: Analysis of Toeplitz Operators. Springer, Berlin (1990)

    Book  Google Scholar 

  11. Brown, A., Halmos, P.R.: Algebraic properties of Toeplitz operators. J. Reine Angew. Math. 213, 89–102 (1963)

    MathSciNet  MATH  Google Scholar 

  12. Cho, G., Yuan, Y.: Bergman metric on the symmetrized bidisc and its consequences. arXiv:2004.04637

  13. Choi, M.D., Effros, E.G.: Injectivity and operator spaces. J. Funct. Anal. 24, 156–209 (1977)

    Article  MathSciNet  Google Scholar 

  14. Didas, M., Eschmeier, J.: Dual Toeplitz operators on the sphere via spherical isometries. Integral Equ. Oper. Theory 83, 291–300 (2015)

    Article  MathSciNet  Google Scholar 

  15. Dullerud, G.E., Paganini, F.: A Course in Robust Control Theory: A Convex Approach. Texts in Applied Mathematics, vol. 36. Springer, New York (2000)

    Book  Google Scholar 

  16. Edigarian, A., Zwonek, W.: Geometry of the symmetrized polydisc. Arch. Math. (Basel) 84, 364–374 (2005)

    Article  MathSciNet  Google Scholar 

  17. Feintuch, A.: On asymptotic Toeplitz and Hankel operators. In: The Gohberg Anniversary Collection, Vol. II (Calgary, AB, 1988), volume 41 of Oper. Theory Adv. Appl., pp. 241–254. Birkhauser, Basel (1989)

  18. Gorai, S., Sarkar, J.: Characterizations of symmetrized polydisc. Indian J. Pure Appl. Math. 47, 391–397 (2016)

    Article  MathSciNet  Google Scholar 

  19. Jewell, N.P.: Toeplitz operators on the Bergman spaces and in several complex variables. Proc. Lond. Math. Soc. 41, 193–216 (1980)

    Article  MathSciNet  Google Scholar 

  20. Maji, A., Sarkar, J., Sarkar, S.: Toeplitz and asymptotic Toeplitz operators on \(H^2({\mathbb{D}}^d)\). Bull. Sci. Math. 146, 33–49 (2018)

    Article  MathSciNet  Google Scholar 

  21. Misra, G., Shyam Roy, S., Zhang, G.: Reproducing kernel for a class of weighted Bergman spaces on the symmetrized polydisc. Proc. Am. Math. Soc. 141, 2361–2370 (2013)

    Article  MathSciNet  Google Scholar 

  22. Nikolov, N.: The symmetrized polydisc cannot be exhausted by domains biholomorphic to convex domains. Ann. Polon. Math. 88, 279–283 (2006)

    Article  MathSciNet  Google Scholar 

  23. Nikolov, N., Pflug, P., Thomas, P.J., Zwonek, W.: Estimates of the Carathéodory metric on the symmetrized polydisc. J. Math. Anal. Appl. 341, 140–148 (2008)

    Article  MathSciNet  Google Scholar 

  24. Nikolov, N., Pflug, P., Zwonek, W.: The Lempert function of the symmetrized polydisc in higher dimensions is not a distance. Proc. Am. Math. Soc. 135, 2921–2928 (2007)

    Article  MathSciNet  Google Scholar 

  25. Nikolov, N., Thomas, P.J., Tran, D.: Lifting maps from the symmetrized polydisc in small dimensions. Complex Anal. Oper. Theory 10, 921–941 (2016)

    Article  MathSciNet  Google Scholar 

  26. Ogle, D.J.: Operator and Function Theory of the Symmetrized Polydisc. Ph.D. thesis, University of Newcastle upon Tyne, October (1999)

  27. Prunaru, B.: Some exact sequences for Toeplitz algebras of spherical isometries. Proc. Am. Math. Soc. 135, 3621–3630 (2007)

    Article  MathSciNet  Google Scholar 

  28. Stroethoff, K., Zheng, D.C.: Toeplitz and Hankel operators on Bergman spaces. Trans. Am. Math. Soc. 329, 773–794 (1992)

    Article  MathSciNet  Google Scholar 

  29. Stroethoff, K., Zheng, D.: Algebraic and spectral properties of dual Toeplitz operators. Trans. Am. Math. Soc. 354(6), 2495–2520 (2002)

    Article  MathSciNet  Google Scholar 

  30. Thomas, P.J., Trao, N.V., Zwonek, W.: Green functions of the spectral ball and symmetrized polydisk. J. Math. Anal. Appl. 377, 624–630 (2011)

    Article  MathSciNet  Google Scholar 

  31. Trybula, M.: Invariant metrics on the symmetrized bidisc. Complex Var. Elliptic Equ. 60, 559–565 (2015)

    Article  MathSciNet  Google Scholar 

  32. Vasilevski, N.L.: Commutative Algebras of Toeplitz Operators on the Bergman Space, Operator Theory: Advances and Applications, 185. Birkhäuser Verlag, Basel (2008). xxx+417 pp

  33. Vukotić, D.: Analytic Toeplitz operators on the Hardy space \(H^p\): a survey. Bull. Belg. Math. Soc. Simon Stevin 10(1), 101–113 (2003)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The research works of the first and second named authors are supported by DST-INSPIRE Faculty Fellowships DST/INSPIRE/04/2015/001094 and DST/INSPIRE/04/2018/002458 respectively. The second named author thanks Indian Institute of Technology, Bombay for a post-doctoral fellowship under which most of this work was done.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India

    B. Krishna Das

  2. Department of Mathematics, Indian Institute of Science Education and Research, Dr Homi Bhabha Road, Pashan, Pune, Maharashtra, 411008, India

    Haripada Sau

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  1. B. Krishna Das
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  2. Haripada Sau
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Correspondence to Haripada Sau.

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Communicated by Vladimir Bolotnikov.

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This article is part of the topical collection “Higher Dimensional Geometric Function Theory and Hypercomplex Analysis” edited by Irene Sabadini, Michael Shapiro and Daniele Struppa.

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Krishna Das, B., Sau, H. Algebraic Properties of Toeplitz Operators on the Symmetrized Polydisk. Complex Anal. Oper. Theory 15, 60 (2021). https://doi.org/10.1007/s11785-021-01108-4

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  • DOI: https://doi.org/10.1007/s11785-021-01108-4

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