The classical Lumer-Phillips theorem characterizes the generators of contraction semigroups. I am looking for a similar characterization or at least a sufficient condition for a family of unbounded, linear operators $A(t)$, $t\geq 0$, to generate a propagator $U(t,s)$, $0\leq s \leq t$, of contractions. I would appreciate any hint where I could find such a result.

Thank you in advance!


2 Answers 2


Pazy's 1983 book on operator semigroups (link to zbMATH)

Pazy, A., Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44. New York etc.: Springer-Verlag. VIII, 279 p. (1983). ZBL0516.47023.

has a some sufficient conditions in the non-autonomous case in Chapter 5. And Section VI.9 in the 2000 book by Engel and Nagel (link to zbMATH)

Engel, Klaus-Jochen; Nagel, Rainer, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p. (2000). ZBL0952.47036.

analyzes the non-autonomous case by rephrasing it as an autonomous problem on a vector-valued function space (but as far as I know, this might often not be overly helpful to tackle concrete PDEs).

It's probably fair to say that, as of today, the non-autonomous situation has still not been understood as well as the autonomous case.

  • $\begingroup$ Thank you for your answer. If I'm not mistaken, in both books they do not cover propagators of contractions. Do you know of any reference for that case? Or is that usually shown directly in the specific case at hand? $\endgroup$ Jul 10, 2023 at 13:41
  • $\begingroup$ @Luke: Are you interested in results where dissipativity of the operators $A(t)$ facilitates the proof of well-posedness (as the Lumer--Phillips theorem that you cited does in the autonomous case) or are you rather interested in situations where you already know well-posedess and want to chracterize when the propagators are contractive? $\endgroup$ Jul 10, 2023 at 14:44
  • $\begingroup$ In my situation, I am given operators $A(t)$ and I want to show the well-posedness as well as that the propagators are contractions. I can probably apply the theorem from Pazy's book and then use a renorming rechnique to obtain the contraction property (still have to spell out the details). I was expecting that a general theory exists for the generation of propagators (analogous to the case of semigroups) that would immediately solve my problem, however, it seems that this case is much more involved than the case of semigroups. $\endgroup$ Jul 10, 2023 at 14:56
  • $\begingroup$ @Luke: I don't have sufficient expertise on the topic to make any definite claims - but it is my understanding that well-posedness is, generally speaking, much more difficult to show in the non-autonomous case, precisely because such nice criteria as Lumer-Phillips break down. But it would probably be better if somebody else with more detailed knowledge of the topic could step in here. $\endgroup$ Jul 10, 2023 at 20:43
  • $\begingroup$ @Luka: But now that I think about it, in case that you're working in Hilbert spaces and the operators $A(t)$ are associated to (maybe elliptic?) sesquilinears from, I think there is a classical result by Lions about well-posedness. I don't know if there's a standard reference for it - maybe Lions' 1961 book (in French; link to zbMATH)? $\endgroup$ Jul 10, 2023 at 21:16

As mentioned in Jochen Glueck's answer, there are results in Pazy's book and in Engel and Nagel's book on the generation of propagators. I also dug a little deeper in the literature and found results on the generation of propagators of contraction in Yosida's Functional Analysis book in Section XIV.4 and in Kato's original paper.

All these results are sufficient conditions for the generation of propagators, and it seems like Jochen Glueck was right when he said that "the non-autonomous situation has still not been understood as well as the autonomous case."

  • $\begingroup$ I am late to the party and apparently you have found something sufficient for your needs, but I would like to mention Section 3 in $L^p$-maximal regularity for non-autonomous evolution equations by Arendt, Chill, Fornaro and Poupaud which, I think, fits your problem description well $\endgroup$
    – Hannes
    Nov 23, 2023 at 8:19
  • $\begingroup$ Even though too late but I would like to mention further referenes that I find very intersting. Kisyński, J.. "Sur les opérateurs de Green des problèmes de Cauchy abstraits." Studia Mathematica 23.3 (1964): 285-328. <eudml.org/doc/217075>. Contrarily to most of the other results, this paper treats operators with time dependant domains. Gregor Nickel's thesis and papers with Rainer Nagel contain some intersting examples that emphasise the difficulties in the non autonomous case. $\endgroup$
    – ahdahmani
    Apr 22 at 12:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.