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

Abstract

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 ∈ L ∞ (λ A × λ B × λ C ) and any X, Y ∈ 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) × S 2 (H), S 2 (H)) of bounded bilinear operators on S 2 (H), and the resulting mappinginto 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 func-. This is a bilinear version of Peller's Theorem characterizing double operator integral mappings S 1 (H) → 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.Résumé. -Considérons trois opérateurs normaux A, B, C sur un espace de Hilbert séparable H ainsi que des mesures spectrales scalairesH)) est une isométrie w * -continue. On montre alors qu'étant donnée une fonction φ ∈ L ∞ (λ A × λ B × λ C ), Γ A,B,C (φ) envoie S 2 (H) × S 2 (H) dans S 1 (H), l'espace des opérateurs à trace sur H, si et seulement si φ vérifie

Central and convolution Herz-Schur multipliers

“…Exploiting the fact that commutative (unital) C * -algebras are simply algebras of continuous functions on compact topological spaces, we identify the central Schur and Herz-Schur multipliers with scalar-valued functions on three and two variables, respectively. This allows us to identify a close link, that seems to have remained unnoticed until now, between central multipliers and the bilinear Schur multipliers into the trace class, introduced and characterised by Coine, Le Merdy and Sukochev in [6] (see also [23]).…”

“…We recall some terminology from [6] that will be used in the sequel. Let ϕ ∈ L ∞ (X × Y × Z) and associate with it a bounded bilinear map…”

“…using the standard identification of this tensor product with the Schur multipliers on X × Y . By standard operator space identifications (see [6] and [23]), we have…”

“…Following the proof of Theorem 3.6, and using the identification Z(A) ∼ = C 0 (Z) and Z(A) ′′ ∼ = L ∞ (Z, m), for some measure space (Z, m), we identify ϕ with an element of L ∞ (Z, m) ⊗ (L ∞ (X) ⊗ w * h L ∞ (Y )). Using [6], we see that there exist an index set I and two families (V i ) i∈I , (W i ) i∈I , where…”

“…A concrete description of these objects, which has found numerous applications thereafter, was given by Grothéndieck in his Resumé [13]. Since then, Schur multipliers have played a significant role in operator theory, the theory of Banach spaces, the theory of operator spaces, and have been linked to perturbation theory through the concept of double operator integrals (see [6,23] and the references therein).…”