Skip to main content
SpringerLink
Log in

Product formulas in the framework of mean ergodic theorems

  • Original Paper
  • Published:
Advances in Operator Theory Aims and scope Submit manuscript

Abstract

An extension of Chernoff’s product formula for one-parameter functions taking values in the space of bounded linear operators on a Banach space is given. Essentially, the n-th one-parameter function in the product formula is mapped by the n-th iterate of a contraction acting on the space of the one-parameter functions. The motivation to study this specific product formula lies in the growing field of dynamical control of quantum systems, involving the procedure of dynamical decoupling and also the Quantum Zeno effect.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arendt, W., Ulm, M.: Trotter’s product formula for projections. Ulmer Seminare, pp. 394–399 (1997)

  2. Arenz, C., Hillier, R., Fraas, M., Burgarth, D.: Distinguishing decoherence from alternative quantum theories by dynamical decoupling. Phys. Rev. A 92(2), 022102 (2015)

    Article  Google Scholar 

  3. Barankai, N., Zimborás, Z.: Generalized quantum Zeno dynamics and ergodic means. arXiv:1811.02509

  4. Bernád, J,Z.: Dynamical control of quantum systems in the context of mean ergodic theorems. J. Phys. A Math. Theor. 50(6), 065303 (2017)

    Article  MathSciNet  Google Scholar 

  5. Chernoff, P.R.: Note on product formulas for operator semigroups. J. Funct. Anal. 2(2), 238–242 (1968)

    Article  MathSciNet  Google Scholar 

  6. Chernoff, P.R.: Semigroup product formulas and addition of unbounded operators. Bull. Am. Math. Soc. 76(2), 395–398 (1970)

    Article  MathSciNet  Google Scholar 

  7. Dunford, N., Schwartz, J.T.: Linear Operators, Part I: General Theory. Interscience Publishers, New York (1964)

    MATH  Google Scholar 

  8. Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)

    MATH  Google Scholar 

  9. Exner, P., Ichinose, T.: A product formula related to quantum zeno dynamics. Ann. H. Poincaré 6(6), 195–215 (2005)

    Article  MathSciNet  Google Scholar 

  10. Facchi, P., Pascazio, S.: Quantum Zeno dynamics: mathematical and physical aspects. J. Phys. A Math. Theor. 4149, 493001 (2008)

    Article  MathSciNet  Google Scholar 

  11. Kato, T.: Remarks on pseudo-resolvents and infinitesimal generators of semi-groups. Proc. Jpn. Acad. 35(8), 467–468 (1959)

    Article  MathSciNet  Google Scholar 

  12. Kato, T., Tanabe, H.: On the abstract evolution equation. Osaka J. Math. 14(1), 107–133 (1962)

    MathSciNet  MATH  Google Scholar 

  13. Krengel, U.: Ergodic Theorems. de Gruyter, Berlin (1985)

    Book  Google Scholar 

  14. Lorch, E.R.: Means of iterated transformations in reflexive vector spaces. Bull. Am. Math. Soc. 45(1), 945–947 (1939)

    Article  MathSciNet  Google Scholar 

  15. Matolcsi, M., Shvidkoy, R.: Trotter’s product formula for projections. Arch. Math. 81(3), 309–317 (2003)

    Article  MathSciNet  Google Scholar 

  16. Möbus, T., Wolf, M.M.: Quantum Zeno effect generalized. J. Math. Phys. 60(5), 052201 (2019)

    Article  MathSciNet  Google Scholar 

  17. Trotter, H.: Approximation of semi-groups of operators. Pac. J. Math. 8(4), 887–919 (1958)

    Article  MathSciNet  Google Scholar 

  18. Trotter, H.: On the product of semigroups of operators. Proc. Am. Math. Soc. 10(4), 545–551 (1959)

    Article  Google Scholar 

  19. Vuillermot, P.-A.: A generalization of Chernoff’s product formula for time-dependent ope-rators. J. Funct. Anal. 259(11), 2923–2938 (2010)

    Article  MathSciNet  Google Scholar 

  20. Vuillermot, P.-A., Wreszinski, W.F., Zagrebnov, V.A.: A general Trotter-Kato formula for a class of evolution operators. J. Funct. Anal. 257(7), 2246–2290 (2009)

    Article  MathSciNet  Google Scholar 

  21. Yosida, K.: Mean ergodic theorem in Banach spaces. Proc. Imp. Acad. Tokyo 14(8), 292–294 (1938)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 732894 (FET Proactive HOT).

Author information

Authors and Affiliations

  1. Department of Physics, University of Malta, Msida, MSD 2080, Malta

    J. Z. Bernád

Authors
  1. J. Z. Bernád
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Z. Bernád.

Additional information

Communicated by Un Cig Ji.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bernád, J.Z. Product formulas in the framework of mean ergodic theorems. Adv. Oper. Theory 5, 15–26 (2020). https://doi.org/10.1007/s43036-019-00001-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s43036-019-00001-4

Keywords

Mathematics Subject Classification

Search

Navigation