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Difference estimates for continuous and discrete operator semigroups

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Abstract

For suitable bounded operator semigroups (e tA) t≥0 in a Banach space, we characterize the estimate ‖Ae tA‖≤c/F(t) for large t, where F is a function satisfying a sublinear growth condition. The characterizations are by holomorphy estimates on the semigroup, and by estimates on powers of the resolvent. We give similar characterizations of the difference estimate ‖T n−T n+1‖≤c/F(n) for a power-bounded linear operator T, when F(n) grows faster than n 1/2 for large n.

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References

  1. Alexopoulos, G.: Sub-Laplacians with drift on Lie groups of polynomial volume growth, Mem. Am. Math. Soc. 155(739) (2002)

  2. Blunck, S.: Maximal regularity of discrete and continuous time evolution equations. Stud. Math. 146, 157–176 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Blunck, S.: Analyticity and discrete maximal regularity on L p -spaces. J. Funct. Anal. 183, 211–230 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Davies, E.B.: One-Parameter Semigroups. London Math. Soc. Monographs, vol. 15. Academic Press, London (1980)

    MATH  Google Scholar 

  5. Dungey, N.: Properties of centered random walks on locally compact groups and Lie groups. Rev. Mat. Iberoam. 23, 587–634 (2007)

    MATH  MathSciNet  Google Scholar 

  6. Dungey, N.: On time regularity and related conditions for power-bounded operators. Preprint, Macquarie University (2007). Accepted for Proceedings of the London Mathematical Society

  7. Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, New York (2000)

    MATH  Google Scholar 

  8. Glynn, P.: Upper bounds on Poisson tail probabilities. Oper. Res. Lett. 6, 9–14 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kalton, N., Montgomery-Smith, S., Oleszkiewicz, K., Tomilov, Y.: Power-bounded operators and related norm estimates. J. Lond. Math. Soc. 70, 463–478 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Nevanlinna, O.: Convergence of Iterations for Linear Equations. Birkhäuser, Basel (1993)

    MATH  Google Scholar 

  11. Nevanlinna, O.: On the growth of the resolvent operators for power bounded operators. In: Janas, J., Szafraniec, F.H., Zemánek, J. (eds.) Linear Operators. Banach Center Publication, vol. 38, pp. 247–264. Institute of Mathematics of the Polish Academy of Sciences, Warsaw (1997)

    Google Scholar 

  12. Nevanlinna, O.: Resolvent conditions and powers of operators. Stud. Math. 145, 113–134 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Yosida, K.: Functional Analysis, 6th edn. Grundlehren der Mathematischen Wissenschaften, vol. 123. Springer, Berlin (1980)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Macquarie University, Sydney, NSW, 2109, Australia

    Nick Dungey

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  1. Nick Dungey
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Correspondence to Nick Dungey.

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Communicated by Rainer Nagel.

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Dungey, N. Difference estimates for continuous and discrete operator semigroups. Semigroup Forum 78, 226–237 (2009). https://doi.org/10.1007/s00233-008-9081-5

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  • DOI: https://doi.org/10.1007/s00233-008-9081-5

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