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Weak $$L^ p$$-stability of a linear semigroup on a Hilbert space implies exponential stability. (English) Zbl 0675.47031
Summary: We prove that a strongly continuous semigroup of linear operators on a Hilbert space is weakly $$L^ p$$-stable for some $$p\in [1,\infty)$$ if and only if the semigroup is exponentially stable. As an application, we prove that the Cauchy problems associated with the semigroup are well posed on the infinite time interval $$(-\infty,0]$$ if and only if the semigroup is exponentially stable.

##### MSC:
 47D03 Groups and semigroups of linear operators 34G10 Linear differential equations in abstract spaces
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##### References:
 [1] Butzer, P.L; Berens, H, Semi-groups of operators and approximation, () · Zbl 0164.43702 [2] Datko, R, Extending a theorem of A. M. Liapunov to Hilbert space, J. math. anal. appl., 32, 610-616, (1970) · Zbl 0211.16802 [3] Diestel, J; Uhl, J.J, Vector measures, A.M.S. mathematical surveys, Vol. 15, (1977), Providence, RI · Zbl 0369.46039 [4] Greiner, G; Voigt, J; Wolff, M, On the spectral bound of the generator of semigroups of positive operators, J. operator theory, 5, 245-256, (1981) · Zbl 0469.47032 [5] Falun, Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. differential equations, 1, 43-56, (1985) · Zbl 0593.34048 [6] () [7] Pazy, A, On the applicability of Lyapunov’s theorem in Hilbert space, SIAM J. math. anal., 3, 291-294, (1972) · Zbl 0242.47028 [8] Pazy, A, Semigroups of linear operators and applications to P.D.E.’s, () · Zbl 0516.47023 [9] Pritchard, A.J; Zabczyk, J, Stability and stabilizability of infinite dimensional systems, SIAM rev., 23, 25-52, (1983) · Zbl 0452.93029 [10] Prüss, J, On the spectrum of C0-semigroups, Trans. amer. math. soc., 284, 847-857, (1984) · Zbl 0572.47030 [11] Przyluski, K.M, On a discrete-time version of a problem of A. J. pritchard and J. zabczyk, () · Zbl 0583.93053 [12] Rudin, W, Real and complex analysis, (1974), McGraw-Hill New York [13] Slemrod, M, Asymptotic behavior of C0 semi-groups as determined by the spectrum of the generator, Indiana univ. math. J., 25, 783-792, (1976) · Zbl 0326.47044 [14] Voigt, J, Interpolation for (positive) C0-semigroups on lp-spaces, Math. Z., 188, 283-286, (1985) · Zbl 0552.47017 [15] \scG. Weiss, The resolvent growth assumption for semigroups on Hilbert spaces, submitted. · Zbl 0693.47034 [16] \scG. Weiss, Weakly l^p-stable linear operators are power stable, submitted. · Zbl 0686.93081
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