An extreme example concerning factorization products on the Schwartz space 𝕾  (R<sup>n</sup>)


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

DOI Code: 10.1285/i15900932v25n2p31

Keywords: Factorization product; Partial algebra

Classification: 47L60; 47A70; 46F99; 47C99

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