An extreme example concerning factorization products on the Schwartz space

Abstract

We construct linear operators S, T mapping the Schwartz space 𝕾 into its dual , such that any operator may be obtained as factorization product . More precisely, given , there exists a Hilbert space such that , the embeddings and are continuous, is dense in , , and S has a continuous extension such that for all φ ∈ 𝕾.