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

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

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

Considérons trois opérateurs normaux $A,B,C$ sur un espace de Hilbert séparable $\mathscr{H}$ ainsi que des mesures spectrales scalaires ${\lambda }_A$ sur $\sigma \left(A\right)$, ${\lambda }_B$ sur $\sigma \left(B\right)$ et ${\lambda }_C$ sur $\sigma \left(C\right)$. Pour tout $\varphi \in L^{\infty }({\lambda }_A\times {\lambda }_B\times {\lambda }_C)$ et pour tous $X,Y\in S^2\left(\mathscr{H}\right)$, l’espace des opérateurs de Hilbert–Schmidt sur $\mathscr{H}$, nous donnons une définition générale d’une intégrale triple d’opérateurs ${\Gamma }^{A,B,C}\left(\varphi \right)(X,Y)$ appartenant à $S^2\left(\mathscr{H}\right)$, de sorte que ${\Gamma }^{A,B,C}\left(\varphi \right)$ appartient à l’espace $B_2(S^2\left(\mathscr{H}\right)\times S^2\left(\mathscr{H}\right),S^2\left(\mathscr{H}\right))$ des opérateurs bilinéaires bornés sur $S^2\left(\mathscr{H}\right)$, et l’application ${\Gamma }^{A,B,C}:L^{\infty }({\lambda }_A\times {\lambda }_B\times {\lambda }_C)\to B_2(S^2\left(\mathscr{H}\right)\times S^2\left(\mathscr{H}\right),S^2\left(\mathscr{H}\right))$ est une isométrie $w^{*}$-continue. On montre alors qu’étant donnée une fonction $\varphi \in L^{\infty }({\lambda }_A\times {\lambda }_B\times {\lambda }_C)$, ${\Gamma }^{A,B,C}\left(\varphi \right)$ envoie $S^2\left(\mathscr{H}\right)\times S^2\left(\mathscr{H}\right)$ dans $S^1\left(\mathscr{H}\right)$, l’espace des opérateurs à trace sur $\mathscr{H}$, si et seulement si $\varphi$ vérifie la propriété de factorisation suivante : il existe un espace de Hilbert $H$ et deux fonctions $a\in L^{\infty }({\lambda }_A\times {\lambda }_B;H)$ et $b\in L^{\infty }({\lambda }_B\times {\lambda }_C;H)$ tels que $\varphi (t_1,t_2,t_3)=〈a(t_1,t_2),b(t_2,t_3)〉$ pour presque tout $(t_1,t_2,t_3)\in \sigma \left(A\right)\times \sigma \left(B\right)\times \sigma \left(C\right)$. Il s’agit de la version bilinéaire du Théorème de Peller caractérisant les applications d’intégrales doubles d’opérateurs envoyant $S^1\left(\mathscr{H}\right)$ dans $S^1\left(\mathscr{H}\right)$. 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 ${\Gamma }_2(E,F^{*})$ 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 $\mathscr{H}$ as well as scalar-valued spectral measures ${\lambda }_A$ on $\sigma \left(A\right)$, ${\lambda }_B$ on $\sigma \left(B\right)$ and ${\lambda }_C$ on $\sigma \left(C\right)$. For any $\varphi \in L^{\infty }({\lambda }_A\times {\lambda }_B\times {\lambda }_C)$ and any $X,Y\in S^2\left(\mathscr{H}\right)$, the space of Hilbert–Schmidt operators on $\mathscr{H}$, we provide a general definition of a triple operator integral ${\Gamma }^{A,B,C}\left(\varphi \right)(X,Y)$ belonging to $S^2\left(\mathscr{H}\right)$ in such a way that ${\Gamma }^{A,B,C}\left(\varphi \right)$ belongs to the space $B_2(S^2\left(\mathscr{H}\right)\times S^2\left(\mathscr{H}\right),S^2\left(\mathscr{H}\right))$ of bounded bilinear operators on $S^2\left(\mathscr{H}\right)$, and the resulting mapping ${\Gamma }^{A,B,C}:L^{\infty }({\lambda }_A\times {\lambda }_B\times {\lambda }_C)\to B_2(S^2\left(\mathscr{H}\right)\times S^2\left(\mathscr{H}\right),S^2\left(\mathscr{H}\right))$ is a $w^{*}$-continuous isometry. Then we show that a function $\varphi \in L^{\infty }({\lambda }_A\times {\lambda }_B\times {\lambda }_C)$ has the property that ${\Gamma }^{A,B,C}\left(\varphi \right)$ maps $S^2\left(\mathscr{H}\right)\times S^2\left(\mathscr{H}\right)$ into $S^1\left(\mathscr{H}\right)$, the space of trace class operators on $\mathscr{H}$, 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 $\varphi (t_1,t_2,t_3)=〈a(t_1,t_2),b(t_2,t_3)〉$ for a.e. $(t_1,t_2,t_3)\in \sigma \left(A\right)\times \sigma \left(B\right)\times \sigma \left(C\right)$. This is a bilinear version of Peller’s Theorem characterizing double operator integral mappings $S^1\left(\mathscr{H}\right)\to S^1\left(\mathscr{H}\right)$. 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.

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

[1] Aleksandrov, Aleksei; Nazarov, Fedor; Peller, Vladimir V. Triple operator integrals in Schatten–von Neumann norms and functions of perturbed noncommuting operators, C. R. Math. Acad. Sci. Paris, Volume 353 (2015) no. 8, pp. 723-728 | DOI | MR | Zbl

[2] Aleksandrov, Aleksei; Nazarov, Fedor; Peller, Vladimir V. Functions of noncommuting self-adjoint operators under perturbation and estimates of triple operator integrals, Adv. Math., Volume 295 (2016), pp. 1-52 | DOI | MR | Zbl

[3] Aleksandrov, Aleksei; Peller, Vladimir V. Multiple operator integrals, Haagerup and Haagerup-like tensor products, and operator ideals, Bull. Lond. Math. Soc., Volume 49 (2017) no. 3, pp. 463-479 | DOI | MR | Zbl

[4] Azamov, Nurulla A.; Carey, Alan L.; Dodds, Peter G.; Sukochev, Fedor Operator integrals, spectral shift, and spectral flow, Can. J. Math., Volume 61 (2009) no. 2, pp. 241-263 | DOI | MR | Zbl

[5] Birman, Mikhail S.; Solomjak, Mikhaĭl Z. Double Stieltjes operator integrals, Probl. Mat. Fiz., Volume 1 (1966), pp. 33-67

[6] Birman, Mikhail S.; Solomjak, Mikhaĭl Z. Double Stieltjes operator integrals. II, Probl. Mat. Fiz., Volume 2 (1967), pp. 26-60

[7] Birman, Mikhail S.; Solomjak, Mikhaĭl Z. Double Stieltjes operator integrals. III. Limit under the integral sign, Probl. Mat. Fiz., Volume 6 (1973), pp. 27-53

[8] Birman, Mikhail S.; Solomjak, Mikhaĭl Z. Double operator integrals in a Hilbert space, Integral Equations Oper. Theory, Volume 47 (2003) no. 2, pp. 131-168 | DOI | MR

[9] Coine, Clément Complete boundedness of multiple operator integrals (2019) (https://arxiv.org/abs/1908.07879, to appear in Canad. Math. Bull.)

[10] Coine, Clément Perturbation theory and higher order $S^p$-differentiability of operator functions (2019) (https://arxiv.org/abs/1906.05585)

[11] Coine, Clément; Le Merdy, Christian; Potapov, Denis; Sukochev, Fedor; Tomskova, Anna Peller’s problem concerning Koplienko–Neidhardt trace formulae: the unitary case, J. Funct. Anal., Volume 271 (2016) no. 7, pp. 1747-1763 | DOI | MR | Zbl

[12] Coine, Clément; Le Merdy, Christian; Potapov, Denis; Sukochev, Fedor; Tomskova, Anna Resolution of Peller’s problem concerning Koplienko–Neidhardt trace formulae, Proc. Lond. Math. Soc., Volume 113 (2016) no. 2, pp. 113-139 | DOI | MR | Zbl

[13] Coine, Clément; Le Merdy, Christian; Skripka, Anna; Sukochev, Fedor Higher order $S^2$-differentiability and application to Koplienko trace formula, J. Funct. Anal., Volume 276 (2019) no. 10, pp. 3170-3204 | DOI | MR | Zbl

[14] Conway, John B. A course in operator theory, Graduate Studies in Mathematics, 21, American Mathematical Society, 2000

[15] Daletskii, Yuriĭ L.; Krein, Selim G. Integration and differentiation of functions of Hermitian operators and applications to the theory of perturbations, Voronež. Gos. Univ. Trudy Sem. Funkcional. Anal., Volume 56 (1956) no. 1, pp. 81-105 | MR

[16] Davidson, Kenneth R. $C^{*}$-algebras by example, Fields Institute Monographs, 6, American Mathematical Society, 1996

[17] Diestel, Joe; Jarchow, Hans; Tonge, Andrew Absolutely summing operators, Cambridge Studies in Advanced Mathematics, 43, Cambridge University Press, 1995 | DOI

[18] Diestel, Joe; Uhl, Jerry J. Vector measures, Mathematical Surveys, 15, American Mathematical Society, 1977 | DOI

[19] Dunford, Nelson; Pettis, Billy J. Linear operations on summable functions, Trans. Am. Math. Soc., Volume 47 (1940), pp. 323-392 | DOI | MR | Zbl

[20] Effros, Edward G.; Ruan, Zhong-Jin Multivariable multipliers for groups and their operator algebras, Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988) (Proceedings of Symposia in Pure Mathematics), Volume 51, American Mathematical Society, 1990, pp. 197-218 | MR | Zbl

[21] Haagerup, Uffe Decomposition of completely bounded maps on operator algebras (1980) (unpublished preprint, Odense University, Denmark)

[22] Hiai, Fumio; Kosaki, Hideki Means of Hilbert space operators, Lecture Notes in Mathematics, 1820, Springer, 2003 | DOI

[23] Jefferies, Brian Singular bilinear integrals in quantum physics, Mathematics, Volume 3 (2015) no. 3, pp. 563-603 | DOI | Zbl

[24] Juschenko, Kate; Todorov, Ivan G.; Turowska, Lyudmila Multidimensional operator multipliers, Trans. Am. Math. Soc., Volume 361 (2009) no. 9, pp. 4683-4720 | DOI | MR | Zbl

[25] Le Merdy, Christian; Skripka, Anna Higher order differentiability of operator functions in Schatten norms, J. Inst. Math. Jussieu, Volume 19 (2020) no. 6, pp. 1993-2016 | DOI | MR | Zbl

[26] Pavlov, Boris S. Multidimensional operator integrals, Probl. Mat. Anal., Volume 2 (1969), pp. 99-122

[27] Pedersen, Gert K. $C^{*}$-algebras and their automorphism groups, London Mathematical Society Monographs, 14, Academic Press Inc., 1979

[28] Peller, Vladimir V. Hankel operators in the theory of perturbations of unitary and selfadjoint operators, Funkts. Anal. Prilozh., Volume 19 (1985) no. 2, pp. 37-51 | MR

[29] Peller, Vladimir V. Multiple operator integrals and higher operator derivatives, J. Funct. Anal., Volume 233 (2006) no. 2, pp. 515-544 | DOI | MR | Zbl

[30] Peller, Vladimir V. Multiple operator integrals in perturbation theory, Bull. Math. Sci., Volume 6 (2016) no. 1, pp. 15-88 | DOI | MR | Zbl

[31] Pisier, Gilles Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conference Series in Mathematics, 60, American Mathematical Society, 1986 | DOI

[32] Pisier, Gilles Similarity problems and completely bounded maps, Lecture Notes in Mathematics, 1618, Springer, 1996 | DOI | MR

[33] Pisier, Gilles Grothendieck’s theorem, past and present, Bull. Am. Math. Soc., Volume 49 (2012) no. 2, pp. 237-323 | DOI | MR | Zbl

[34] Potapov, Denis; Skripka, Anna; Sukochev, Fedor Spectral shift function of higher order, Invent. Math., Volume 193 (2013) no. 3, pp. 501-538 | DOI | MR | Zbl

[35] Potapov, Denis; Skripka, Anna; Sukochev, Fedor; Tomskova, Anna Multilinear Schur multipliers and Schatten properties of operator Taylor remainders, Adv. Math., Volume 320 (2017), pp. 1063-1098 | DOI | MR | Zbl

[36] Reed, Michael; Simon, Barry Methods of modern mathematical physics. I Functional analysis, Academic Press Inc., 1980

[37] Rudin, Walter Functional analysis, McGraw-Hill, 1973 (McGraw-Hill Series in Higher Mathematics)

[38] Skripka, Anna; Tomskova, Anna Multilinear operator integrals, Lecture Notes in Mathematics, 2250, Springer, 2019 (Theory and applications) | DOI

[39] Spronk, Nico Measurable Schur multipliers and completely bounded multipliers of the Fourier algebras, Proc. Lond. Math. Soc., Volume 89 (2004) no. 1, pp. 161-192 | DOI | MR | Zbl