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1996 | 121 | 2 | 167-183
Tytuł artykułu

A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let T be a semigroup of linear contractions on a Banach space X, and let $X_{s}(T) = {x ∈ X : lim_{s→∞} ∥T(s)x∥ = 0}$. Then $X_{s}(T)$ is the annihilator of the bounded trajectories of T*. If the unitary spectrum of T is countable, then $X_{s}(T)$ is the annihilator of the unitary eigenvectors of T*, and $lim_{s} ∥T(s)x∥ = inf{∥x-y∥ : y ∈ X_{s}(T)}$ for each x in X.
Czasopismo
Rocznik
Tom
121
Numer
2
Strony
167-183
Opis fizyczny
Daty
wydano
1996
otrzymano
1996-01-15
poprawiono
1996-07-15
Twórcy
  • 87 Eversley Road, Benfleet, Essex SS7 4JT, England
Bibliografia
  • [1] W. Arendt, Gaussian estimates and interpolation of the spectrum in $L^p$, Differential Integral Equations 7 (1994), 1153-1168.
  • [2] W. Arendt and C. J. K. Batty, Tauberian theorms and stability of one-parameter semigroups, Trans. Amer. Math. Soc. 306 (1988), 837-852.
  • [3] E. Balslev, The essential spectrum of elliptic differential operators in $L^p(ℝ_n)$, ibid. 116 (1965), 193-217.
  • [4] C. J. K. Batty, Asymptotic stability of Schrödinger semigroups: path integral methods, Math. Ann. 292 (1992), 457-492.
  • [5] C. J. K. Batty and D. A. Greenfield, On the invertibility of isometric semigroup representations, Studia Math. 110 (1994), 235-250.
  • [6] C. J. K. Batty and Vũ Quôc Phóng, Stability of strongly continuous representations of abelian semigroups, Math. Z. 209 (1992), 75-88.
  • [7] Z. Brze/xniak and B. Szafirski, Asymptotic behaviour of $L^1$ norm of solutions to parabolic equations, Bull. Polish Acad. Sci. Math. 39 (1991), 1-10.
  • [8] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, Cambridge, 1989.
  • [9] E. B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995), 141-169.
  • [10] E. B. Davies, Long time asymptotics of fourth order parabolic equations, J. Anal. Math., to appear.
  • [11] R. G. Douglas, On extending commutative semigroups of operators, Bull. London Math. Soc. 1 (1969), 157-159.
  • [12] J. Esterle, E. Strouse et F. Zouakia, Stabilité asymptotique de certains semigroupes d'opérateurs et idéaux primaires de $L^1(ℝ^+)$, J. Operator Theory 28 (1992), 203-227.
  • [13] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., 1964.
  • [14] D. Gilbarg and J. Serrin, On isolated singularities of solutions of second order elliptic differential equations, J. Anal. Math. 4 (1955-56), 309-340.
  • [15] J.-P. Kahane et Y. Katznelson, Sur les algèbres de restrictions des séries de Taylor absolument convergentes à un fermé du cercle, ibid. 23 (1970), 185-197.
  • [16] Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), 313-328.
  • [17] L. H. Loomis, The spectral characterization of a class of almost periodic functions, Ann. of Math. 72 (1960), 362-368.
  • [18] Yu. I. Lyubich, Introduction to the Theory of Banach Representations of Groups, Birkhäuser, Basel, 1988.
  • [19] Yu. I. Lyubich and Vũ Quôc Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math. 88 (1988), 37-42.
  • [20] Yu. I. Lyubich and Vũ Quôc Phóng, A spectral criterion for the almost periodicity of one-parameter semigroups, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 47 (1987), 36-41 (in Russian).
  • [21] Yu. I. Lyubich and Vũ Quôc Phóng, A spectral criterion for asymptotic almost periodicity of uniformly continuous representations of abelian semigroups, ibid. 50 (1988), 38-43 (in Russian); English transl.: J. Soviet Math. 49 (1990), 1263-1266.
  • [22] Z. M. Ma and M. Röckner, An Introduction to the Theory of Non-Symmetric Dirichlet Forms, Springer, Berlin, 1992.
  • [23] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577-591.
  • [24] R. Nagel (ed.), One-Parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer, Berlin, 1986.
  • [25] J. van Neerven, The Asymptotic Behaviour of a Semigroup of Linear Operators, Birkhäuser, Basel, 1996.
  • [26] E. M. Ouhabaz, $L^∞$-contractivity of semigroups generated by sectorial forms, J. London Math. Soc. 46 (1992), 529-542.
  • [27] R. Hempel and J. Voigt, The spectrum of a Schrödinger operator in $L_p(ℝ^ν)$ is p-independent, Comm. Math. Phys. 104 (1986), 243-250.
  • [28] G. K. Pedersen, C*-Algebras and their Automorphism Groups, Academic Press, London, 1979.
  • [29] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer, Berlin, 1984.
  • [30] D. W. Robinson, Elliptic Operators and Lie Groups, Oxford Univ. Press, Oxford, 1991.
  • [31] W. Rudin, Fourier Analysis on Groups, Wiley, New York, 1992.
  • [32] R. Rudnicki, Asymptotic stability in $L^1$ of parabolic equations, J. Differential Equations 102 (1993), 391-401.
  • [33] M. Schechter, Spectra of Partial Differential Operators, North-Holland, Amsterdam, 1971.
  • [34] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447-526.
  • [35] Vũ Quôc Phóng, Theorems of Katznelson-Tzafriri type for semigroups of operators, J. Funct. Anal. 103 (1992), 74-84.
  • [36] Vũ Quôc Phóng, On the spectrum, complete trajectories, and asymptotic stability of linear semi-dynamical systems, J. Differential Equations 105 (1993), 30-45.
  • [37] Vũ Quôc Phóng, Stability and almost periodicity of trajectories of periodic processes, ibid. 115 (1995), 402-415.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv121i2p167bwm
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