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C*-algebra generated by horizontal Toeplitz operators on the Fock space

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Abstract

We introduce the so-called horizontal Toeplitz operators acting on the Fock space and give an explicit description of the C*-algebra generated by them. We show that any Toeplitz operator with \(L_{\infty }\)-symbol, which is invariant under imaginary translations, is unitarily equivalent to the multiplication operator by its “spectral function”. This result is also true for the Toeplitz operators whose defining symbols are invariant under translations over any Lagrangian plane. The main result of the paper states that the corresponding spectral functions form a dense subset in the C*-algebra of bounded uniformly continuous functions with respect to the standard metric on \(\mathbb {R}^{n}\).

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References

  1. Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. Commun. Pure Appl. Math. 3, 187–214 (1961)

  2. Berezin, F.A.: General concept of quantization. Commun. Math. Phys. 40, 153–174 (1975)

  3. Berger, C.A., Coburn, L.A.: Toeplitz operators and quantum mechanics. J. Funct. Anal. 68, 273–299 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. de Gosson, M.A.: Symplectic methods in harmonic analysis and in mathematical physics. Pseudo-Differential Operators: Theory and Applications, vol. 7 Birkhäuser, Basel (2011)

  5. Esmeral, K., Maximenko, E.A.: \(C^{*}\)-algebra of angular Toeplitz operators on Bergman spaces over the upper half-plane. Commun. Math. Anal. 17(2), 151–162 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Esmeral, K., Maximenko, E.A.: Radial Toeplitz operators on the Fock space and square-root-slowly oscillating sequences. Complex Equ. Oper. Theory, Submitted (2015)

  7. Esmeral, K., Maximenko, E.A., Vasilevski, N.: \(C^{*}\)-algebra generated by angular Toeplitz operators on the weighted Bergman spaces over the upper half-plane. Integr. Equ. Oper. Theory, to appear (2015)

  8. Fock, V.A.: Konfigurationsraum und zweite Quantelung. Z. Phys. 75, 622–647 (1932)

    Article  MATH  Google Scholar 

  9. Folland, G.B.: A course in Abstract Harmonic Analysis. Studies in Advanced Mathematics, CRC Press, Boca Raton (1995)

  10. Folland, G.B.: Harmonic Analysis in Phase Space. Princeton University Press (1989)

  11. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals. Series and Product, Academic Press (1980)

  12. Grudsky, S., Karapetyants, A., Vasilevski, N.: Dynamics of properties of Toeplitz operators with radial symbols. Integr. Equ. Oper. Theory 20, 217–253 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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)

    MathSciNet  MATH  Google Scholar 

  14. Grudsky, S., Karapetyants, A., Vasilevski, N.: Dynamics of properties of Toeplitz operators on the upper half-plane: Hyperbolic case. Bol. Soc. Mat. Mex. (3rd series) 10, 119–138 (2004)

  15. Grudsky, S., Quiroga-Barranco, R., Vasilevski, N.L.: Commutative C*-algebras of Toeplitz operators and quantization on the unit disk. J. Funct. Anal. 234, 1–44 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grudsky, S., 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 

  17. Grudsky, S., Vasilevski, N.: Toeplitz operators on the Fock space: radial component effects. Integr. Equ. Oper. Theory 44(1), 10–37 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Herrera Yañez, C., Hutník, O., Maximenko, E.A.: Vertical symbols, Toeplitz operators on weighted Bergman spaces over the upper half-plane and very slowly oscillating functions. C. R. Acad. Sci. Paris, Ser. I 352, 129–132 (2014)

  19. Herrera Yañez, C., Maximenko, E.A., Vasilevski, N.L.: Vertical Toeplitz operators on the upper half-plane and very slowly oscillating functions. Integr. Equ. Oper. Theory 77, 149–166 (2013)

  20. Reiter, H., Stegeman, J.D.: Classical Harmonic Analysis and Locally Compact Groups, 2nd edn. Clarendon Press, Oxford (2000)

    MATH  Google Scholar 

  21. Segal, I.E.: Lectures at the summer seminar on Applied Math. Boulder, Colorado (1960)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Vasilevski, N.L.: Commutative Algebras of Toeplitz Operators on the Bergman Space. Operator Theory: Advances and Applications, vol. 185. Birkhäuser, Basel (2008)

  24. Zhu, K.: Analysis on Fock Spaces. Graduate Texts in Mathematics, vol. 263. Springer, New York (2012)

  25. Zhu, K.: Operator Theory in Function Spaces. Mathematical surveys and monographs, vol 138, 2nd Edition. American Mathematical Society, Rhode Island (2007)

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Authors and Affiliations

  1. Department of Mathematics of the Center for Research and Advanced Studies of the National Polytechnic Institute (CINVESTAV), Mexico city, Mexico

    Kevin Esmeral & Nikolai Vasilevski

Authors
  1. Kevin Esmeral
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  2. Nikolai Vasilevski
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Corresponding author

Correspondence to Nikolai Vasilevski.

Additional information

To Sergei Grudsky on the occasion of his 60th anniversary.

This work was partially supported by CONACYT Project 238630 México.

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Esmeral, K., Vasilevski, N. C*-algebra generated by horizontal Toeplitz operators on the Fock space . Bol. Soc. Mat. Mex. 22, 567–582 (2016). https://doi.org/10.1007/s40590-016-0110-1

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  • DOI: https://doi.org/10.1007/s40590-016-0110-1

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