A $C_0$-semigroup is a strongly continuous family $\{T(t)\}$ of bounded linear operators on a normed space $X$, indexed in $\mathbb R_+$ and with two additional properties that make it look like an operator-valued version of the exponential function: $T(t)T(s)=T(t+s)$ for all $t,s\in \mathbb R_+$ and $T(0)=\hbox{Id}_X$. On the basis of two concrete examples I was wondering whether there is room to generalise this notions to operator families indexed on general directed sets. For this purpose, defining a sum-like operation on a directed set seems to be unavoidable and I was wondering whether this has been done in the past.

(I admit that the only two examples I can think of are rather trivial: $\mathbb R$ viewed as a collection of two halflines directed towards 0 (which returns the definition of $C_0$-group), as well as the positive hyperoctant in $\mathbb R^n$ (which for $n=2$ reminds of how analytic $C_0$-semigroups are defined). But what I am striving for is the definition of $C_0$-semigroups indexed on some kind of tree-like sets.)

  • $\begingroup$ It is helpful to mention that the research of $C_0$ semigroups is largely motivated by the study of wellposedness of parabolic and hyperbolic PDEs (see Section 7.4 of Evan's PDE book). In this setting it is natural to restrict the time domain to $\mathbb R_{\ge0}$, though it is conceivable to generalize it to some cone-like regions in $\mathbb R^n$. $\endgroup$ – Fan Zheng Dec 14 '15 at 15:21
  • $\begingroup$ Sure. I was actually playing with the idea of some kind of ramified timespace, although I was thinking of some evolutionary model/philogenetic time-tree rather than of multiverses or things like that. But the question seems to be well-defined on a pure theoretical level already. $\endgroup$ – Delio Mugnolo Dec 14 '15 at 15:24
  • $\begingroup$ However, I must add that I am surprised that the answers have come only from analysts so far. I feel that what I am asking for really is a question about algebraic structures (defining a monoid on a directed set), I feel there might be a reason if nobody has ever considered something like that. $\endgroup$ – Delio Mugnolo Dec 15 '15 at 7:41

This does not answer your question only adds something to your list of examples: two parameter semigroups, as investigated by

Slocinski, M. Unitary dilation of two-parameter semi-groups of contractions. (English) [J] Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 22, 1011-1014 (1974)

and many others later on. They are significantly different from analytic semigroups defined on a sector of the complex plane.

ADDED: An other possible generalization is the concept of a strongly continuous cocycle, see Definition 6.1 in

Carmen Chicone,Yuri Latushkin: Evolution Semigroups in Dynamical Systems and Differential Equations, American Mathematical Soc., 1999.

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  • $\begingroup$ Thanks for your references, András. The definiton in the book of Chicone/Latushkin seems promising, even if the cocycle does not satisfy the classical exponential law; but the idea might work in some special cases at least, including my tree-like generalisation of time. $\endgroup$ – Delio Mugnolo Dec 15 '15 at 7:47

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