On stability of 𝐶₀-semigroups

Phong, Vu

doi:10.1090/S0002-9939-01-05614-3

On stability of $C_0$-semigroups
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by Vu Quoc Phong

Proc. Amer. Math. Soc. 129 (2001), 2871-2879

DOI: https://doi.org/10.1090/S0002-9939-01-05614-3

Published electronically: May 10, 2001

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Abstract:

We prove that if $T(t)$ is a $C_0$-semigroup on a Hilbert space $E$, then (a) $1\in \rho (T(\omega ))$ if and only if $\sup \{\|\int ^t_0\exp \{(2\pi ik)/\omega \}T(s)x ds\|\colon \ t\geq 0, k\in \mathbf {Z}\}<\infty$, for all $x\in E$, and (b) $T(t)$ is exponentially stable if and only if $\sup \{\|\int ^t_0\exp \{i\lambda t\}T(s)x ds\|\colon \ t\geq 0, \lambda \in \mathbf {R}\}<\infty$, for all $x\in E$. Analogous, but weaker, statements also hold for semigroups on Banach spaces.

References

Charles J. K. Batty and Vũ Quốc Phóng, Stability of individual elements under one-parameter semigroups, Trans. Amer. Math. Soc. 322 (1990), no. 2, 805–818. MR 1022866, DOI 10.1090/S0002-9947-1990-1022866-5
Larry Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc. 236 (1978), 385–394. MR 461206, DOI 10.1090/S0002-9947-1978-0461206-1
I. Herbst, The spectrum of Hilbert space semigroups, J. Operator Theory 10 (1983), no. 1, 87–94. MR 715559
James S. Howland, On a theorem of Gearhart, Integral Equations Operator Theory 7 (1984), no. 1, 138–142. MR 802373, DOI 10.1007/BF01204917
T. F. McCabe, A note on iterates that are contractions, J. Math. Anal. Appl. 104 (1984), no. 1, 64–66. MR 765039, DOI 10.1016/0022-247X(84)90030-1
W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck, One-parameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184, Springer-Verlag, Berlin, 1986. MR 839450, DOI 10.1007/BFb0074922
J. M. A. M. van Neerven, Individual stability of $C_0$-semigroups with uniformly bounded local resolvent, Semigroup Forum 53 (1996), no. 2, 155–161. MR 1400641, DOI 10.1007/BF02574130
Jan van Neerven, The asymptotic behaviour of semigroups of linear operators, Operator Theory: Advances and Applications, vol. 88, Birkhäuser Verlag, Basel, 1996. MR 1409370, DOI 10.1007/978-3-0348-9206-3
J. M. A. M. van Neerven, Characterization of exponential stability of a semigroup of operators in terms of its action by convolution on vector-valued function spaces over $\mathbf R_+$, J. Differential Equations 124 (1996), no. 2, 324–342. MR 1370144, DOI 10.1006/jdeq.1996.0012
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
Jan Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc. 284 (1984), no. 2, 847–857. MR 743749, DOI 10.1090/S0002-9947-1984-0743749-9
Vũ Quôc Phóng, On the exponential stability and dichotomy of $C_0$-semigroups, Studia Math. 132 (1999), no. 2, 141–149. MR 1669694, DOI 10.4064/sm-132-2-141-149
Quoc Phong Vu and E. Schüler, The operator equation $AX-XB=C$, admissibility, and asymptotic behavior of differential equations, J. Differential Equations 145 (1998), no. 2, 394–419. MR 1621042, DOI 10.1006/jdeq.1998.3418
George Weiss, Weak $L^p$-stability of a linear semigroup on a Hilbert space implies exponential stability, J. Differential Equations 76 (1988), no. 2, 269–285. MR 969425, DOI 10.1016/0022-0396(88)90075-7
George Weiss, Weakly $l^p$-stable linear operators are power stable, Internat. J. Systems Sci. 20 (1989), no. 11, 2323–2328. MR 1031158, DOI 10.1080/00207728908910309