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

Coine, Clément; Le Merdy, Christian; Sukochev, Fedor

doi:10.5802/aif.3422

When do triple operator integrals take value in the trace class?
[Quand les intégrales triples d’opérateurs sont-elles à valeurs dans les opérateurs à trace ?]

Coine, Clément ¹ ; Le Merdy, Christian ¹ ; Sukochev, Fedor ²

¹ Laboratoire de Mathématiques de Besançon UMR 6623 CNRS, Université Bourgogne Franche-Comté 25030 Besançon Cedex (France)
² School of Mathematics & Statistics University of NSW Kensington NSW 2052 (Australia)

Annales de l'Institut Fourier, Tome 71 (2021) no. 4, pp. 1393-1448.

Résumé
Abstract

Considérons trois opérateurs normaux $A, B, C$ sur un espace de Hilbert séparable $ℋ$ ainsi que des mesures spectrales scalaires $λ_{A}$ sur $σ (A)$ , $λ_{B}$ sur $σ (B)$ et $λ_{C}$ sur $σ (C)$ . Pour tout $ϕ \in L^{\infty} (λ_{A} \times λ_{B} \times λ_{C})$ et pour tous $X, Y \in S^{2} (ℋ)$ , l’espace des opérateurs de Hilbert–Schmidt sur $ℋ$ , nous donnons une définition générale d’une intégrale triple d’opérateurs $Γ^{A, B, C} (ϕ) (X, Y)$ appartenant à $S^{2} (ℋ)$ , de sorte que $Γ^{A, B, C} (ϕ)$ appartient à l’espace $B_{2} (S^{2} (ℋ) \times S^{2} (ℋ), S^{2} (ℋ))$ des opérateurs bilinéaires bornés sur $S^{2} (ℋ)$ , et l’application $Γ^{A, B, C} : L^{\infty} (λ_{A} \times λ_{B} \times λ_{C}) \to B_{2} (S^{2} (ℋ) \times S^{2} (ℋ), S^{2} (ℋ))$ est une isométrie $w^{*}$ -continue. On montre alors qu’étant donnée une fonction $ϕ \in L^{\infty} (λ_{A} \times λ_{B} \times λ_{C})$ , $Γ^{A, B, C} (ϕ)$ envoie $S^{2} (ℋ) \times S^{2} (ℋ)$ dans $S^{1} (ℋ)$ , l’espace des opérateurs à trace sur $ℋ$ , si et seulement si $ϕ$ vérifie la propriété de factorisation suivante : il existe un espace de Hilbert $H$ et deux fonctions $a \in L^{\infty} (λ_{A} \times λ_{B}; H)$ et $b \in L^{\infty} (λ_{B} \times λ_{C}; H)$ tels que $ϕ (t_{1}, t_{2}, t_{3}) = 〈a (t_{1}, t_{2}), b (t_{2}, t_{3})〉$ pour presque tout $(t_{1}, t_{2}, t_{3}) \in σ (A) \times σ (B) \times σ (C)$ . Il s’agit de la version bilinéaire du Théorème de Peller caractérisant les applications d’intégrales doubles d’opérateurs envoyant $S^{1} (ℋ)$ dans $S^{1} (ℋ)$ . On établit en passant qu’étant donnés deux espaces de Banach séparables $E$ et $F$ , toute fonction $w^{*}$ -mesurable et essentiellement bornée à valeurs dans l’espace $Γ_{2} (E, F^{*})$ des opérateurs de $E$ dans $F^{*}$ se factorisant par un espace de Hilbert, admet une factorisation hilbertienne $w^{*}$ -mesurable.

Consider three normal operators $A, B, C$ on a 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 $Γ^{A, B, C} : L^{\infty} (λ_{A} \times λ_{B} \times λ_{C}) \to B_{2} (S^{2} (ℋ) \times S^{2} (ℋ), S^{2} (ℋ))$ 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 $ϕ (t_{1}, t_{2}, t_{3}) = 〈a (t_{1}, t_{2}), b (t_{2}, t_{3})〉$ 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.

Reçu le : 2019-01-25
Révisé le : 2020-04-08
Accepté le : 2020-05-13
Première publication : 2021-12-08
Publié le : 2022-03-15

DOI : 10.5802/aif.3422

Classification : 47B10, 47B38, 46E40
Keywords: Trace class, Triple operator integrals, Schur multipliers, Factorization through Hilbert space
Mot clés : Opérateurs à trace, Intégrales triples d’opérateurs, Multiplicateurs de Schur, Factorisation par un espace de Hilbert

Affiliations des auteurs :

Coine, Clément ¹ ; Le Merdy, Christian ¹ ; Sukochev, Fedor ²

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {When do triple operator integrals take value in the trace class?},
     journal = {Annales de l'Institut Fourier},
     pages = {1393--1448},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Coine, Clément; Le Merdy, Christian; Sukochev, Fedor. When do triple operator integrals take value in the trace class?. Annales de l'Institut Fourier, Tome 71 (2021) no. 4, pp. 1393-1448. doi : 10.5802/aif.3422. https://aif.centre-mersenne.org/articles/10.5802/aif.3422/

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