2017/06/06 by Clément Coine, Coine, Clément, Christian Le Merdy +3 · 2 citations
Mathematics · #46E40 #47B10 #47B38 #Advanced Harmonic Analysis Research #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory
paper · doi:10.48550/arxiv.1706.01662
openalex publication_date 2017/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider three normal operators A,B,C on separable Hilbert space \H as well as scalar-valued spectral measures λA on σ(A), λB on σ(B) and λC on σ(C). For any ϕ∈ L^∞(λA× λB× λC) and any X,Y∈ S2(\H), the space of Hilbert-Schmidt operators on \H, we provide a general definition of a triple operator integral ΓA,B,C(ϕ)(X,Y) belonging to S2(\H) in such a way that ΓA,B,C(ϕ) belongs to the space B2(S2(\H)× S2(\H), S2(\H)) of bounded bilinear operators on S2(\H), and the resulting mapping ΓA,B,C\colon L^∞(λA× λB× λC) → B2(S2(\H)× S2(\H), S2(\H)) is a w^*-continuous isometry. Then we show that a function ϕ∈ L^∞(λA× λB× λC) has the property that ΓA,B,C(ϕ) maps S2(\H)× S2(\H) into S1(\H), the space of trace class operators on \H, if and only if it has the following factorization property: there exist a Hilbert space H and two functions a∈ L∞(λA × λB ; H) and b∈ L∞(λB× λC ; H) such that ϕ(t1,t2,t3)= ⟨ a(t1,t2),b(t2,t3) ⟩ for a.e. (t1,t2,t3) ∈ σ(A) × σ(B) × σ(C). This is a bilinear version of Peller's Theorem characterizing double operator integral mappings S1(\H)→ S1(\H). In passing we show that for any separable Banach spaces E,F, any w^*-measurable esssentially bounded function valued in the Banach space Γ2(E,F^*) of operators from E into F^* factoring through Hilbert space admits a w^*-measurable Hilbert space factorization.