Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations:¹⁻²

Let $f$ be analytic and consider the one-dimensional (the multidimensional case is more intricate, but can be done in a similar fashion³) differential equation 
$$ \dot{x} = f(x), \ x(0)=x_0. \tag{*}\label{*} $$

By defining the variables 
$$ \phi_{k} = x^{k},\  k=1,2,\ldots$$

one has that \eqref{*} can be written as an infinite dimensional linear system given by 
$$ \dot{\Phi} = A {\Phi}, $$

where ${\Phi} =(\phi_1,\phi_2,\ldots)$.

**Does every operator $A$ produced by Carleman linearization generates a strongly continuous semigroup?**

(1): <cite cite="_Acta Math._ **59** (1932), no. 1, 63--87" mrnumber="1555355" authors="Torsten Carleman">_Torsten Carleman_, MR 1555355 [**Application de la théorie des équations intégrales linéaires aux systèmes d’équations différentielles non linéaires**](http://dx.doi.org/10.1007/BF02546499), _Acta Math._ **59** (1932), no. 1, 63--87.</cite>

(2): <cite cite="_IMA J. Math. Control Info._ **13** (1996), no. 3, 279--298" authors="D. McCaffrey, S. P. Banks, and A. Moser">_D. McCaffrey, S. P. Banks and A. Moser_, [**Clifford algebras, dynamical systems, and periodic orbits**](http://dx.doi.org/10.1093/imamci/13.3.279), _IMA J. Math. Control Info._ **13** (1996), no. 3, 279--298.</cite>

(3): _Joaquín Collado and Irving Sánchez_, [**Modified Carleman Linearization and its use in oscillators**](http://dx.doi.org/10.1109/ICEEE.2008.4723445), _Proc. of 5th Int. Conf. Elec. Eng. Comp. Sci. Aut. Control_ (2008), 13--19.