×

Semigroups of sums of two operators with small commutators. (English) Zbl 1514.47067

Summary: Let \(A\) be the generator of a \(C_0\)-semigroup \((e^{At})_{t\geq0}\) on a Banach space \(\mathcal{X}\) and \(B\) be a bounded operator in \(\mathcal{X}\). Assuming that \(\int_{0}^{\infty}\|e^{At}\|\|e^{Bt}\|\,dt<\infty\) and the commutator \(AB-BA\) is bounded and has a sufficiently small norm, we show that \(\int_{0}^{\infty}\|e^{(A+B)t}\|\,dt<\infty\), where \((e^{(A+B)t})_{t\geq0}\) is the semigroup generated by \(A+B\). In addition, estimates for the supremum- and \(L^1\)-norms of the difference \(e^{(A+B)t}-e^{At}e^{Bt}\) are derived.

MSC:

47D06 One-parameter semigroups and linear evolution equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adler, M., Bombieri, M., Engel, K.-J.: On perturbations of generators of \[C_0\] C0-semigroups. Abstr. Appl. Anal. (2014). https://doi.org/10.1155/2014/213020 · doi:10.1155/2014/213020
[2] Batty, C.J.K.: On a perturbation theorem of Kaiser and Weis. Semigroup Forum 70, 471-474 (2005) · Zbl 1098.47035 · doi:10.1007/s00233-005-0504-2
[3] Batty, C.J.K., Krol, S.: Perturbations of generators of \[C_0\] C0-semigroups and resolvent decay. J. Math. Anal. Appl. 367, 434-443 (2010) · Zbl 1198.47056 · doi:10.1016/j.jmaa.2010.01.048
[4] Buse, C., Khan, A., Rahmat, G., Saierli, O.: Weak real integral characterizations for exponential stability of semigroups in reflexive spaces. Semigroup Forum 88, 195-204 (2014) · Zbl 1393.47019 · doi:10.1007/s00233-013-9520-9
[5] Buse, C., Niculescu, C.: A condition of uniform exponential stability for semigroups. Math. Inequal. Appl. 11(3), 529-536 (2008) · Zbl 1180.47027
[6] Eisner, T.: Stability of Operators and Operator Semigroups, Operator Theory: Advances and Applications, vol. 209. Birkhä, Basel (2010) · Zbl 1205.47002
[7] Gil’, M.I.: Operator Functions and Localization of Spectra. Lecture Notes in Mathematics, vol. 1830. Springer, Berlin (2003) · Zbl 1032.47001 · doi:10.1007/b93845
[8] Guo, B., Zwart, H.: On the relation between stability of continuous – and discrete-time evolution equations via the Cayley transform. Integral Equ. Oper. Theory 54, 349-383 (2006) · Zbl 1104.34040 · doi:10.1007/s00020-003-1350-9
[9] Hadd, S.: Unbounded perturbations of \[C_0\] C0-semigroups on Banach spaces and applications. Semigroup Forum 70(3), 451-465 (2005) · Zbl 1074.47017 · doi:10.1007/s00233-004-0172-7
[10] Heymann, R.: Eigenvalues and stability properties of multiplication operators and multiplication semigroups. Math. Nachr. 287(5-6), 574-584 (2014) · Zbl 1524.47048 · doi:10.1002/mana.201300046
[11] Matrai, T.: On perturbations preserving the immediate norm continuity of semigroups. J. Math. Anal. Appl. 341, 961-974 (2008) · Zbl 1138.47035 · doi:10.1016/j.jmaa.2007.10.048
[12] Paunonen, L., Zwart, H.: A Lyapunov approach to strong stability of semigroups. Syst. Control Lett. 62, 673-678 (2013) · Zbl 1279.93083 · doi:10.1016/j.sysconle.2013.05.001
[13] Seifert, C., Wingert, D.: On the perturbation of positive semigroups. Semigroup Forum 91, 495-501 (2015) · Zbl 1385.47016 · doi:10.1007/s00233-014-9651-7
[14] Weiss, G.: Weak \[L_p\] Lp-stability of linear semigroup on a Hilbert space implies exponential stability. J. Differ. Equ. 76, 269-285 (1988) · Zbl 0675.47031 · doi:10.1016/0022-0396(88)90075-7
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.