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Individual stability of \(C_0\)-semigroups with uniformly bounded local resolvent. (English) Zbl 0892.47040
In the spirit of L. Gearhart’s theorem [Trans. Am. Math. Soc. 236, 385-394 (1978; Zbl 0371.47033)] the author proves the following individual stability theorem for strongly continuous semigroups \((T(t))\) on a Banach space X with generator A. If, for some \(x_0 \in X\), the map \(z \mapsto R(z,A)x_0\) has a bounded analytic extension to \(\{z:\text{Re}z > 0\}\) then \(\| T(t)R(\lambda, A)x_0\| \leq M(1 + t)\) for all \(t\geq 0\), some (all) \(\lambda \in \varrho(A)\) and some \(M \in \mathbb R_+\). As a corollary he obtains the recent theorem by L. Weis and V. Wrobel [Proc. Am. Math. Soc. 124, No. 12, 3663-3671 (1996; Zbl 0863.47027)].

47D06 One-parameter semigroups and linear evolution equations
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[1] Arendt, W., and Batty, C.J.K.,Tauberian theorems and stability of one-parameter semigroups, Trans. Am. Math. Soc.306 (1988), 837–852. · Zbl 0652.47022 · doi:10.1090/S0002-9947-1988-0933321-3
[2] Gearhart, L.,Spectral theory for contraction semigroups on Hilbert spaces, Trans. Am. Math. Soc.236 (1978), 385–394. · Zbl 0326.47038 · doi:10.1090/S0002-9947-1978-0461206-1
[3] Greiner, G., Voigt, J., and Wolff, M.P.H.,On the spectral bound of the generator of positive semigroups, J. Operator Th.5 (1981), 245–256. · Zbl 0469.47032
[4] Hille, E., and Phillips, R.S., ”Functional Analysis and Semi-Groups”, Coll. Publ. Am. Math. Soc. XXXI, Providence, R.I., 1957. · Zbl 0078.10004
[5] Huang Falun,Exponential stability of linear systems in Banach spaces, Chin. Ann. Math.10B (1989), 332–340. · Zbl 0694.47027
[6] Nagel, R. (ed.), ”One-parameter Semigroups of Positive Operators”, Springer Lect. Notes in Math. 1184, Springer-Verlag, 1986. · Zbl 0585.47030
[7] Neerven, J.M.A.M. van, Characterization of exponential stability of a semigroup of operators in terms of its action by convolution on vector-valued function spaces over \(\mathbb{R}\)+, to appear in: J. Diff. Eq. · Zbl 0913.47033
[8] Neerven, J.M.A.M. van, Straub, B., and Weis, L.,on the asymptotic behaviour of a semigroup of linear operators, submitted.
[9] Neubrander, F., ”Well-posedness of higher order abstract Cauchy problems”, Ph.D. Dissertation, University of Tübingen, 1984. · Zbl 0542.34053
[10] Neubrander, F.,Laplace transform and asymptotic behavior of strongly continuous semigroups, Houston J. Math.12 (1986), 549–561. · Zbl 0624.47031
[11] Slemrod, M.,Asymptotic behavior of C 0-semigroups as determined by the spectrum of the generator, Indiana Univ. Math. J.25 (1976), 783–892. · Zbl 0326.47044 · doi:10.1512/iumj.1976.25.25062
[12] Weis, L., and Wrobel, V.,Asymptotic behavior of C 0-semigroups in Banach spaces, preprint. · Zbl 0863.47027
[13] Weiss, G.,The resolvent growth assumption for semigroups on Hilbert spaces, J. Math. Anal. Appl.145 (1990), 154–171. · Zbl 0693.47034 · doi:10.1016/0022-247X(90)90438-L
[14] Wrobel, V.,Asymptotic behavior of C 0-semigroups in B-convex Banach spaces, Indiana Univ. Math. J.38 (1989), 101–114. · Zbl 0653.47018 · doi:10.1512/iumj.1989.38.38004
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