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): Torsten Carleman, MR 1555355 Application de la théorie des équations intégrales linéaires aux systèmes d’équations différentielles non linéaires, Acta Math. 59 (1932), no. 1, 63--87.
(2): D. McCaffrey, S. P. Banks and A. Moser, Clifford algebras, dynamical systems, and periodic orbits, IMA J. Math. Control Info. 13 (1996), no. 3, 279--298.
(3): Joaquín Collado and Irving Sánchez, Modified Carleman Linearization and its use in oscillators, Proc. of 5th Int. Conf. Elec. Eng. Comp. Sci. Aut. Control (2008), 13--19.
