Norm continuity of c 0 -semigroups

V. Goersmeyer; L. Weis

Studia Mathematica (1999)

  • Volume: 134, Issue: 2, page 169-178
  • ISSN: 0039-3223

Abstract

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We show that a positive semigroup T t on L p ( Ω , ν ) with generator A and ||R(α + i β)|| → 0 as |β| → ∞ for some α ∈ ℝ is continuous in the operator norm for t>0. The proof is based on a criterion for norm continuity in terms of “smoothing properties” of certain convolution operators on general Banach spaces and an extrapolation result for the L p -scale, which may be of independent interest.

How to cite

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Goersmeyer, V., and Weis, L.. "Norm continuity of $c_0$-semigroups." Studia Mathematica 134.2 (1999): 169-178. <http://eudml.org/doc/216630>.

@article{Goersmeyer1999,
abstract = {We show that a positive semigroup $T_t$ on $L_p(Ω,ν)$ with generator A and ||R(α + i β)|| → 0 as |β| → ∞ for some α ∈ ℝ is continuous in the operator norm for t>0. The proof is based on a criterion for norm continuity in terms of “smoothing properties” of certain convolution operators on general Banach spaces and an extrapolation result for the $L_p$-scale, which may be of independent interest.},
author = {Goersmeyer, V., Weis, L.},
journal = {Studia Mathematica},
keywords = {convolution operators; norm continuity of a -semigroup; spaces of -integrable functions; asymptotic behaviour of the resolvent of the generator},
language = {eng},
number = {2},
pages = {169-178},
title = {Norm continuity of $c_0$-semigroups},
url = {http://eudml.org/doc/216630},
volume = {134},
year = {1999},
}

TY - JOUR
AU - Goersmeyer, V.
AU - Weis, L.
TI - Norm continuity of $c_0$-semigroups
JO - Studia Mathematica
PY - 1999
VL - 134
IS - 2
SP - 169
EP - 178
AB - We show that a positive semigroup $T_t$ on $L_p(Ω,ν)$ with generator A and ||R(α + i β)|| → 0 as |β| → ∞ for some α ∈ ℝ is continuous in the operator norm for t>0. The proof is based on a criterion for norm continuity in terms of “smoothing properties” of certain convolution operators on general Banach spaces and an extrapolation result for the $L_p$-scale, which may be of independent interest.
LA - eng
KW - convolution operators; norm continuity of a -semigroup; spaces of -integrable functions; asymptotic behaviour of the resolvent of the generator
UR - http://eudml.org/doc/216630
ER -

References

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  1. [1] H. Alt, Lineare Funktionalanalysis, Springer, Berlin, 1992. 
  2. [2] J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976. Zbl0344.46071
  3. [3] E. B. Davies, One-Parameter Semigroups, Academic Press, London, 1980. Zbl0457.47030
  4. [4] A. Driouich and O. El-Mennaoui, On the inverse formula of Laplace transforms, preprint. 
  5. [5] O. El-Mennaoui and K. J. Engel, On the characterization of eventually norm continuous semigroups in Hilbert space. Arc. Math. (Basel) 63 (1994), 437-440. Zbl0811.47035
  6. [6] O. El-Mennaoui and K. J. Engel, Towards a characterization of eventually norm continuous semigroups on Banach spaces, Quaestiones Math. 19 (1996), 183-190. Zbl0863.47022
  7. [7] J. Martinez and J. Mazon, C 0 -semigroups norm continuous at infinity, Semigroup Forum 52 (1996), 213-224. Zbl0927.47029
  8. [8] J. van Neerven, The Asymptotic Behaviour of Semigroups of Linear Operators, Birkhäuser, Basel, 1996. Zbl0905.47001
  9. [9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. 
  10. [10] L. Weis, A short proof for the stability theorem for positive semigroups on L p ( ν ) , Proc. Amer. Math. Soc. (1998), to appear. Zbl0904.47028
  11. [11] L. Weis, Integral operators and changes of density, Indiana Univ. Math. J. 31 (1982), 83-96. Zbl0492.47017
  12. [12] P. You, Characteristic conditions for c 0 -semigroups with continuity in the uniform operator topology for t>0, Proc. Amer. Math. Soc. 116 (1992), 991-997. Zbl0773.47023

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