Norm continuity of $c_0$-semigroups

• Autorzy: V. Goersmeyer , L. Weis
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 47D06: One-parameter semigroups and linear evolution equations
• 47B65: Positive operators and order-bounded operators
• 47A10: Spectrum, resolvent

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1999
• Tom: 134
• Numer: 2
• Strony: 169-178

Daty

wydano

1999

otrzymano

1998-05-06

poprawiono

1998-10-23

Twórcy

autor

V. Goersmeyer

• Mathematisches Institut I, Universität Karlsruhe, Englerstr. 2, D-76128 Karlsruhe, Germany,

autor

L. Weis

• Mathematisches Institut I, Universität Karlsruhe, Englerstr. 2, D-76128 Karlsruhe, Germany,

Bibliografia

• [1] H. Alt, Lineare Funktionalanalysis, Springer, Berlin, 1992.
• [2] J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976.
• [3] E. B. Davies, One-Parameter Semigroups, Academic Press, London, 1980.
• [4] A. Driouich and O. El-Mennaoui, On the inverse formula of Laplace transforms, preprint.
• [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.
• [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.
• [7] J. Martinez and J. Mazon, $C_0$-semigroups norm continuous at infinity, Semigroup Forum 52 (1996), 213-224.
• [8] J. van Neerven, The Asymptotic Behaviour of Semigroups of Linear Operators, Birkhäuser, Basel, 1996.
• [9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
• [10] L. Weis, A short proof for the stability theorem for positive semigroups on $L_p(ν)$, Proc. Amer. Math. Soc. (1998), to appear.
• [11] L. Weis, Integral operators and changes of density, Indiana Univ. Math. J. 31 (1982), 83-96.
• [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.