zbMATH — the first resource for mathematics

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.

47D03 Groups and semigroups of linear operators
34G10 Linear differential equations in abstract spaces
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.