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 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 mapping) has the property that Γ A,B,C (φ) maps S 2 (H)×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. 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.

Weak $(1,1)$ estimates for multiple operator integrals and generalized absolute value functions

“…In this paper H is an infinite dimensional separable Hilbert space, B(H) is the algebra of all bounded operators on H and B(H) sa stands for the set of all bounded self-adjoint operators. Note that the separability of H is used in [CLS17]. We write Tr for the trace on B(H).…”

“…) and the elementary tensor products are weak- * dense in this space. In [CLS17] it is explained that also the space of bounded multi-linear maps S 2 ×. .…”

Higher order S2-differentiability and application to Koplienko trace formula

“…In this section, we recall a multiple operator integral introduced in [18] and developed in [8] and derive its key algebraic properties that underline our main results. We note that there are several other different constructions of multiple operator integrals [3,4,10,21,24], but they do not suit the generality of this paper because they are applicable to smaller sets of symbols.…”

“…It means that λ A is a positive finite measure on the Borel subsets of σ(A) such that λ A and E A , the spectral measure of A, have the same sets of measure zero. We refer to [9,Section 15] and [8,Section 2.1] for the existence and the construction of such measure.…”

“…. , n, for derivatives ϕ (k) via multiple operator integrals defined in [8], where f [k] is the kth order divided difference of f . The latter representation for ϕ (k) plays a key role in proving existence of ϕ (k+1) (induction process) and in the study of the operator Taylor remainder, which satisfies…”