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In functional analysis, the multiple operator integral is a multilinear map $$T^{H_0,\ldots,H_n}_\phi$$ informally written as
 * $$T^{H_0,\ldots,H_n}_\phi(V_1,\ldots,V_n)=\int\cdots\int \phi(\lambda_0,\ldots,\lambda_n)dE_{H_0}(\lambda_0)V_1 dE_{H_1}(\lambda_1)\cdots V_n dE_{H_n}(\lambda_n),$$

an expression which can be made precise in several different ways. Multiple operator integrals are of use in various situations where functional calculus appears alongside noncommuting operators (e.g., matrices), for instance in perturbation theory, harmonic analysis, index theory, noncommutative geometry, and operator theory in general. As noncommuting operators, functional calculus, and perturbation theory are central to quantum theory, multiple operator integrals are also frequently applied there. Closely related concepts are Schur multiplication and the Feynman operational calculus. Multiple operator integrals were introduced by Peller as multilinear generalizations of double operator integrals, developed by Daletski, Krein, Birman, and Solomyak.

Definition
A conceptually clean definition of the multiple operator integral is given as follows (it is a special case of both and ). Let $$n\in\N$$, let $$\mathcal{H}$$ be a separable Hilbert space, and denote the space of bounded operators by $$\mathcal{B}(\mathcal{H})$$. Let $$H_0,\ldots,H_n$$ be possibly unbounded self-adjoint operators in $$\mathcal{H}$$. For any function $$\phi:\R^{n+1}\to\R$$ (called the symbol) which admits a decomposition
 * $$\phi(\lambda_0,\ldots,\lambda_n)=\int_\Omega a_0(\lambda_0,\omega)\cdots a_1(\lambda_n,\omega)\,{\rm d}\omega$$

for a certain finite measure space $$(\Omega,{\rm d}\omega)$$ and bounded measurable functions $$a_0,\ldots,a_n:\R\times\Omega\to\C$$, the multiple operator integral is the $n$-multilinear operator
 * $$T_\phi^{H_0,\ldots,H_n}:\mathcal{B}(\mathcal{H})\times\cdots\times\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$$

defined by
 * $$T_\phi^{H_0,\ldots,H_n}(V_1,\ldots,V_n)\psi:=\int_\Omega a_0(H_0,\omega)V_1a_1(H_1,\omega)\cdots V_n a_n(H_n,\omega)\psi\,{\rm d}\omega$$

for all $$\psi\in\mathcal{H}$$. One can show that the integrand is Bochner integrable, and (using Banach-Steinhaus) that $$T_\phi^{H_0,\ldots,H_n}$$ is a bounded multilinear map. Moreover, $$T_\phi^{H_0,\ldots,H_n}$$ only depends on $$(\Omega,{\rm d}\omega)$$ and $$a_0,\ldots,a_n$$ through $$\phi$$, as the notation $$T_\phi^{H_0,\ldots,H_n}$$ suggests.

Other definitions
One may similarly define $$T_\phi^{H_0,\ldots,H_n}$$ on the product of Schatten classes $$\mathcal{S}^{p_1}\times\cdots\times\mathcal{S}^{p_n}$$ and end up with a mapping
 * $$T_\phi^{H_0,\ldots,H_n}:\mathcal{S}^{p_1}\times\cdots\times\mathcal{S}^{p_n}\to\mathcal{S}^{p}$$

where $$\frac{1}{p}=\frac{1}{p_1}+\ldots+\frac{1}{p_n}$$. The restriction of the domain allows the multiple operator integral to be defined for a larger class of symbols $$\phi:\R^{n+1}\to\R$$. Because one can (and often needs to) trade of assumptions on $$\phi$$, $$H_0,\ldots,H_n$$, and $$V_1,\ldots,V_n$$, there are several definitions of the multiple operator integral which are not generalizations of one another, but typically agree in the cases where both are defined.

The multiple operator integral can be defined on the product of noncommutative L^p-spaces as
 * $$T_\phi^{H_0,\ldots,H_n}:\mathcal{L}^{p_1}(\mathcal{N},\tau)\times\cdots\times\mathcal{L}^{p_n}(\mathcal{N},\tau)\to\mathcal{L}^p(\mathcal{N},\tau)$$

for a von Neumann algebra $$\mathcal{N}$$ admitting a semifinite trace $$\tau$$. One then additionally assumes that $$H_0,\ldots,H_n$$ are affiliated to $$\mathcal{N}$$.

Divided differences
The most often used symbol of a multiple operator integral is the divided difference $$\phi=f^{[n]}$$ of an $$n$$ times continuously differentiable function $$f:\R\to\R$$, defined recursively as
 * $$f^{[0]}(\lambda_0):=f(\lambda_0)$$
 * $$f^{[n]}(\lambda_0,\ldots,\lambda_n)=\lim_{x\in\R\setminus\{\lambda_0\},x\to\lambda_n}\frac{f^{[n-1]}(\lambda_0,\ldots,\lambda_{n-1})-f^{[n-1]}(\lambda_1,\ldots,\lambda_{n-1},x)}{\lambda_0-x}.$$

In particular, $$f^{[n]}(\lambda,\ldots,\lambda)=\frac{1}{n!}f^{(n)}(\lambda)$$, and
 * $$f^{[1]}(\lambda,\mu)=\frac{f(\lambda)-f(\mu)}{\lambda-\mu}.$$

The multiple operator integral $$T_{f^{[n]}}:\mathcal{B}(\mathcal{H})^{\times n}\to \mathcal{B}(\mathcal{H})$$ is known to exist in the case that $$f$$ in a suitable Besov space, for example, when $$\widehat{f^{(n)}}\in L^1(\R)$$, and the multiple operator integral $$\mathcal{S}^{p_1}\times\cdots\times\mathcal{S}^{p_n}\to\mathcal{S}^p$$ for H\"older conjugate $$p_1,\ldots,p_n,p\in(1,\infty)$$ (as above), is known to exist when $$f\in C^n(\R)$$ with $$f^{(n)}$$ bounded.

Properties
Proofs of the following facts can be found in

Algebraic properties
The double operator integral has the following properties: Using the fact that the multiple operator integral of zero order is simply functional calculus:
 * 1) $$f(A)-f(B)=T^{A,B}_{f^{[1]}}(A-B)$$
 * 2) $$[f(A),B]=T_{f^{[1]}}^{A,A}([A,B])$$
 * $$f(A)=T_{f^{[0]}}^A,$$

one recognizes that 1. and 2. are identities relating multiple operator integrals of 0 order (single operator integrals) to multiple operator integrals of 1st order (double operator integrals). The properties 1. and 2. can be generalized as follows
 * 1) $$T_{f^{[n]}}^{H_0,\ldots,H_{j-1},A,H_{j+1},\ldots,H_n}(V_1,\ldots,V_n)-T_{f^{[n]}}^{H_0,\ldots,H_{j-1},B,H_{j+1},\ldots,H_n}(V_1,\ldots,V_n)=T^{H_0,\ldots,H_{j-1},A,B,H_{j+1},\ldots,H_n}_{f^{[n+1]}}(V_1,\ldots,V_{j},A-B,V_{j+1},\ldots,V_n)$$
 * 2) $$T_{f^{[n]}}^{H_0,\ldots,H_n} (V_1,\ldots,V_j,a V_{j+1},\ldots,V_n)-T_{f^{[n]}}^{H_0,\ldots,H_n}(V_1,\ldots,V_j a,V_{j+1},\ldots,V_n)=T_{f^{[n+1]}}^{H_0,\ldots,H_n}(V_1,\ldots,V_j,[H_j,a],V_{j+1},\ldots,V_n)$$

In combination with the operator trace $${\rm Tr}$$ (or any other tracial function) the multiple operator integral satisfies the following cyclicity property:
 * $${\rm Tr} (V_0 T_{f^{[n]}}^{H_0,\ldots,H_n} (V_1,\ldots,V_n))={\rm Tr} (V_1 T_{f^{[n]}}^{H_1,\ldots,H_n,H_0} (V_2,\ldots,V_n,V_0))$$

Under suitable conditions, the above identities follow from elementary properties of the divided difference, combined with the fact that $$T_\phi$$ is independent of the integral representation of $$\phi$$.

In quantum theory
$\Tr(e^{-\beta H})$