Compact perturbations of linear differential equations in locally convex spaces

S. A. Shkarin¹

¹ Mathematics Department King's College London, Strand London WC2R 2LS, UK

Studia Mathematica, Tome 172 (2006) no. 3, pp. 203-227

Résumé

Herzog and Lemmert have proven that if $E$ is a Fréchet space and $T : E \to E$ is a continuous linear operator, then solvability (in $[0,1]$) of the Cauchy problem $\dot x=Tx$, $x(0)=x_0$ for any $x_0\in E$ implies solvability of the problem $\dot x(t)=Tx(t)+f(t,x(t))$, $x(0)=x_0$ for any $x_0\in E$ and any continuous map $f:[0,1]\times E\to E$ with relatively compact image. We prove the same theorem for a large class of locally convex spaces including: $\bullet$ DFS-spaces, i.e., strong duals of Fréchet–Schwartz spaces, in particular the spaces of Schwartz distributions $\S'(\mathbb R^n)$, the spaces of distributions with compact support ${\cal E}'({\mit\Omega} )$ and the spaces of germs of holomorphic functions $H(K)$ over an arbitrary compact set $K\subset\mathbb C^n$;$\bullet$ complete LFS-spaces, i.e., complete inductive limits of sequences of Fréchet–Schwartz spaces, in particular the spaces ${\cal D}({\mit\Omega} )$ of test functions;$\bullet$ PLS-spaces, i.e., projective limits of sequences of DFS-spaces, in particular, the spaces ${\cal D}'({\mit\Omega} )$ of distibutions and ${\cal A}({\mit\Omega} )$ of real-analytic functions. Here ${\mit\Omega} $ is an arbitrary open domain in $\mathbb R^n$. We construct an example showing that the analogous statement for (smoothly) time-dependent linear operators is invalid already for Fréchet spaces.

Détail

DOI : 10.4064/sm172-3-1

Keywords: herzog lemmert have proven chet space continuous linear operator solvability cauchy problem dot implies solvability problem dot t continuous map times relatively compact image prove theorem large class locally convex spaces including bullet dfs spaces strong duals chet schwartz spaces particular spaces schwartz distributions mathbb spaces distributions compact support cal mit omega spaces germs holomorphic functions arbitrary compact set subset mathbb bullet complete lfs spaces complete inductive limits sequences chet schwartz spaces particular spaces cal mit omega test functions bullet pls spaces projective limits sequences dfs spaces particular spaces cal mit omega distibutions cal mit omega real analytic functions here mit omega arbitrary domain mathbb construct example showing analogous statement smoothly time dependent linear operators invalid already chet spaces