An extreme example concerning factorization products on the Schwartz space $? (Rⁿ)$

Abstract

We construct linear operators S, T mapping the Schwartz space ? into its dual $?'$, such that any operator $R ∈ ?(?, ?')$ may be obtained as factorization product $S ○ T$. More precisely, given $R ∈ ?(?, ?')$, there exists a Hilbert space $H_R$ such that $? ⊂ H_R ⊂ ?'$, the embeddings $? ↪ H_R$ and $ H_R ↪ ?'$ are continuous, $?$ is dense in $ H_R$, $T(?) ⊂ H_R$, and S has a continuous extension $\widetilde{S} :H_R → ?'$ such that $\widetilde{S}(T φ)=R φ$ for all φ ∈ ?.