When do triple operator integrals take value in the trace class?

Consider three normal operators $A,B,C$ on separable Hilbert space $\H$ as well as scalar-valued spectral measures $λ_A$ on $σ(A)$, $λ_B$ on $σ(B)$ and $λ_C$ on $σ(C)$. For any $ϕ\in L^\infty(λ_A\times λ_B\times λ_C)$ and any $X,Y\in S^2(\H)$, the space of Hilbert-Schmidt operators on $\H$, we provide a general definition of a triple operator integral $Γ^{A,B,C}(ϕ)(X,Y)$ belonging to $S^2(\H)$ in such a way that $Γ^{A,B,C}(ϕ)$ belongs to the space $B_2(S^2(\H)\times S^2(\H), S^2(\H))$ of bounded bilinear operators on $S^2(\H)$, and the resulting mapping $Γ^{A,B,C}\colon L^\infty(λ_A\times λ_B\times λ_C) \to B_2(S^2(\H)\times S^2(\H), S^2(\H))$ is a $w^*$-continuous isometry. Then we show that a function $ϕ\in L^\infty(λ_A\times λ_B\times λ_C)$ has the property that $Γ^{A,B,C}(ϕ)$ maps $S^2(\H)\times S^2(\H)$ into $S^1(\H)$, 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 $a\in L^{\infty}(λ_A \times λ_B ; H)$ and $b\in L^{\infty}(λ_B\times λ_C ; H)$ such that $ϕ(t_1,t_2,t_3)= \left\langle a(t_1,t_2),b(t_2,t_3) \right\rangle$ for a.e. $(t_1,t_2,t_3) \in σ(A) \times σ(B) \times σ(C).$ This is a bilinear version of Peller's Theorem characterizing double operator integral mappings $S^1(\H)\to S^1(\H)$. In passing we show that for any separable Banach spaces $E,F$, any $w^*$-measurable esssentially bounded function valued in the Banach space $Γ_2(E,F^*)$ of operators from $E$ into $F^*$ factoring through Hilbert space admits a $w^*$-measurable Hilbert space factorization.

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