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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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https://hal.archives-ouvertes.fr/hal-03525881
Contributor : Christian Le Merdy Connect in order to contact the contributor
Submitted on : Friday, January 14, 2022 - 10:27:35 AM
Last modification on : Wednesday, January 19, 2022 - 4:37:00 PM
Long-term archiving on: : Friday, April 15, 2022 - 6:28:53 PM

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  • HAL Id : hal-03525881, version 1
  • ARXIV : 1706.01662

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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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