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

Abstract : Consider three normal operators $A,B,C$ on separable Hilbert space $\H$ as well as scalar-valued spectral measures $\lambda_A$ on $\sigma(A)$, $\lambda_B$ on $\sigma(B)$ and $\lambda_C$ on $\sigma(C)$. For any $\phi\in L^\infty(\lambda_A\times \lambda_B\times \lambda_C)$ 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 $\Gamma^{A,B,C}(\phi)(X,Y)$ belonging to $S^2(\H)$ in such a way that $\Gamma^{A,B,C}(\phi)$ 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 $\Gamma^{A,B,C}\colon L^\infty(\lambda_A\times \lambda_B\times \lambda_C) \to B_2(S^2(\H)\times S^2(\H), S^2(\H))$ is a $w^*$-continuous isometry. Then we show that a function $\phi\in L^\infty(\lambda_A\times \lambda_B\times \lambda_C)$ has the property that $\Gamma^{A,B,C}(\phi)$ 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 $a\in L^{\infty}(\lambda_A \times \lambda_B ; H)$ and $b\in L^{\infty}(\lambda_B\times \lambda_C ; H)$ such that $\phi(t_1,t_2,t_3)= \left\langle a(t_1,t_2),b(t_2,t_3) \right\rangle$ for a.e. $(t_1,t_2,t_3) \in \sigma(A) \times \sigma(B) \times \sigma(C).$ 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 $\Gamma_2(E,F^*)$ of operators from $E$ into $F^*$ factoring through Hilbert space admits a $w^*$-measurable Hilbert space factorization.
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Journal articles

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Submitted on : Friday, January 14, 2022 - 10:27:35 AM
Last modification on : Wednesday, January 19, 2022 - 4:37:00 PM
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• HAL Id : hal-03525881, version 1
• ARXIV : 1706.01662

Citation

Clément Coine, Christian Le Merdy, Fedor Sukochev. When do triple operator integrals take value in the trace class?. Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, inPress. ⟨hal-03525881⟩

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