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In functional analysis, the multiple operator integral is a multilinear map ${\displaystyle T_{\phi }^{H_{0},\ldots ,H_{n}}}$ determined by linear operators ${\displaystyle H_{0},\ldots ,H_{n}}$ and a multivariable function ${\displaystyle \phi }$ that lives on the product of the spectra of ${\displaystyle H_{0},\ldots ,H_{n}}$. Multiple operator integrals are of use in various situations where functional calculus appears alongside noncommuting operators (such as matrices or differential operators), 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 in their modern form were introduced by Peller.^[1] They are multilinear generalizations of double operator integrals, developed by Daletski, Krein, Birman, and Solomyak, among others.

A conceptually clean definition of the multiple operator integral is given as follows. Let ${\displaystyle n\in \mathbb {N} }$, let ${\displaystyle {\mathcal {H}}}$ be a separable Hilbert space, and denote the space of bounded operators by ${\displaystyle {\mathcal {B}}({\mathcal {H}})}$. Let ${\displaystyle H_{0},\ldots ,H_{n}}$ be possibly unbounded self-adjoint operators in ${\displaystyle {\mathcal {H}}}$. For any function ${\displaystyle \phi :\mathbb {R} ^{n+1}\to \mathbb {R} }$ (called the symbol) which admits a decomposition

${\displaystyle \phi (\lambda _{0},\ldots ,\lambda _{n})=\int _{\Omega }a_{0}(\lambda _{0},\omega )\cdots a_{1}(\lambda _{n},\omega )\,{\rm {d}}\omega }$

for a certain finite measure space ${\displaystyle (\Omega ,{\rm {d}}\omega )}$ and bounded measurable functions ${\displaystyle a_{0},\ldots ,a_{n}:\mathbb {R} \times \Omega \to \mathbb {C} }$, the multiple operator integral is the ${\displaystyle n}$-multilinear operator

${\displaystyle T_{\phi }^{H_{0},\ldots ,H_{n}}:{\mathcal {B}}({\mathcal {H}})\times \cdots \times {\mathcal {B}}({\mathcal {H}})\to {\mathcal {B}}({\mathcal {H}})}$

defined by

${\displaystyle T_{\phi }^{H_{0},\ldots ,H_{n}}(V_{1},\ldots ,V_{n})\psi :=\int _{\Omega }a_{0}(H_{0},\omega )V_{1}a_{1}(H_{1},\omega )\cdots V_{n}a_{n}(H_{n},\omega )\psi \,{\rm {d}}\omega }$

for all ${\displaystyle \psi \in {\mathcal {H}}}$. One can show that the integrand is Bochner integrable, and (using Banach-Steinhaus) that ${\displaystyle T_{\phi }^{H_{0},\ldots ,H_{n}}}$ is a bounded multilinear map. Moreover, ${\displaystyle T_{\phi }^{H_{0},\ldots ,H_{n}}}$ only depends on ${\displaystyle (\Omega ,{\rm {d}}\omega )}$ and ${\displaystyle a_{0},\ldots ,a_{n}}$ through ${\displaystyle \phi }$, as the notation ${\displaystyle T_{\phi }^{H_{0},\ldots ,H_{n}}}$ suggests.

One may similarly define ${\displaystyle T_{\phi }^{H_{0},\ldots ,H_{n}}}$ on the product of Schatten classes ${\displaystyle {\mathcal {S}}^{p_{1}}\times \cdots \times {\mathcal {S}}^{p_{n}}}$ and end up with a mapping

${\displaystyle T_{\phi }^{H_{0},\ldots ,H_{n}}:{\mathcal {S}}^{p_{1}}\times \cdots \times {\mathcal {S}}^{p_{n}}\to {\mathcal {S}}^{p}}$

where ${\displaystyle {\frac {1}{p}}={\frac {1}{p_{1}}}+\ldots +{\frac {1}{p_{n}}}}$. The restriction of the domain allows the multiple operator integral to be defined for a larger class of symbols ${\displaystyle \phi :\mathbb {R} ^{n+1}\to \mathbb {R} }$. Because one can (and often needs to) trade of assumptions on ${\displaystyle \phi }$, ${\displaystyle H_{0},\ldots ,H_{n}}$, and ${\displaystyle V_{1},\ldots ,V_{n}}$, 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. In a way all definitions are ways to make precise the formal expression

${\displaystyle T_{\phi }^{H_{0},\ldots ,H_{n}}(V_{1},\ldots ,V_{n})=\int \cdots \int \phi (\lambda _{0},\ldots ,\lambda _{n})dE_{H_{0}}(\lambda _{0})V_{1}dE_{H_{1}}(\lambda _{1})\cdots V_{n}dE_{H_{n}}(\lambda _{n}).}$

The multiple operator integral can also be defined^[2] on the product of noncommutative ${\displaystyle L^{p}}$-spaces as

${\displaystyle T_{\phi }^{H_{0},\ldots ,H_{n}}:{\mathcal {L}}^{p_{1}}({\mathcal {N}},\tau )\times \cdots \times {\mathcal {L}}^{p_{n}}({\mathcal {N}},\tau )\to {\mathcal {L}}^{p}({\mathcal {N}},\tau )}$

for a von Neumann algebra ${\displaystyle {\mathcal {N}}}$ admitting a semifinite trace ${\displaystyle \tau }$. One then additionally assumes that ${\displaystyle H_{0},\ldots ,H_{n}}$ are affiliated to ${\displaystyle {\mathcal {N}}}$.

The most often used symbol of a multiple operator integral is the divided difference ${\displaystyle \phi =f^{[n]}}$ of an ${\displaystyle n}$ times continuously differentiable function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$, defined recursively as

${\displaystyle f^{[0]}(\lambda _{0}):=f(\lambda _{0})}$

${\displaystyle f^{[n]}(\lambda _{0},\ldots ,\lambda _{n}):=\lim _{x\in \mathbb {R} \setminus \{\lambda _{0}\},x\to \lambda _{n}}{\frac {f^{[n-1]}(\lambda _{0},\ldots ,\lambda _{n-1})-f^{[n-1]}(\lambda _{1},\ldots ,\lambda _{n-1},x)}{\lambda _{0}-x}}.}$

It follows that ${\displaystyle f^{[n]}(\lambda _{1},\ldots ,\lambda _{n})=f^{[n]}(\lambda _{\sigma (1)},\ldots ,\lambda _{\sigma (n)})}$ for any permutation ${\displaystyle \sigma }$, ${\displaystyle f^{[n]}(\lambda ,\ldots ,\lambda )={\frac {1}{n!}}f^{(n)}(\lambda )}$, and, for ${\displaystyle \lambda \neq \mu }$,

${\displaystyle f^{[1]}(\lambda ,\mu )={\frac {f(\lambda )-f(\mu )}{\lambda -\mu }}.}$

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

The fact that ${\displaystyle T_{\phi }^{H_{0},\ldots ,H_{n}}}$ does not depend on the integral representation of ${\displaystyle \phi }$ allows one to deduce MOI-identities from the respective identities of the symbols. For example, ${\displaystyle \phi _{1}(x,y,z)=\phi _{2}(x,y,z)(z-i)^{-1}}$ implies ${\displaystyle T_{\phi _{1}}^{H_{0},H_{1},H_{2}}(V_{1},V_{2})=T_{\phi _{2}}^{H_{0},H_{1},H_{2}}(V_{1},V_{2})(H_{2}-i)^{-1}}$ (the right-hand side moreover equals ${\displaystyle T_{\phi _{2}}^{H_{0},H_{1},H_{2}}(V_{1},V_{2}(H_{2}-i)^{-1})}$ by definition of the MOI). One can thus derive operator-theoretic formulas from elementary computations with scalar-valued functions, similar to (but more general than) using the integral kernel of an integral operator, as ubiquitous in quantum field theory. This technique in particular yields useful identities when working with divided differences as symbols.

The double operator integral has the following properties:

1. ${\displaystyle f(A)-f(B)=T_{f^{[1]}}^{A,B}(A-B)}$
2. ${\displaystyle [f(A),B]=T_{f^{[1]}}^{A,A}([A,B])}$

Using the fact that the multiple operator integral of zero order is simply functional calculus:

${\displaystyle f(A)=T_{f^{[0]}}^{A}(),}$

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. ${\displaystyle T_{f^{[n]}}^{H_{0},\ldots ,H_{j-1},A,H_{j+1},\ldots ,H_{n}}(V_{1},\ldots ,V_{n})-T_{f^{[n]}}^{H_{0},\ldots ,H_{j-1},B,H_{j+1},\ldots ,H_{n}}(V_{1},\ldots ,V_{n})=T_{f^{[n+1]}}^{H_{0},\ldots ,H_{j-1},A,B,H_{j+1},\ldots ,H_{n}}(V_{1},\ldots ,V_{j},A-B,V_{j+1},\ldots ,V_{n})}$
2. ${\displaystyle T_{f^{[n]}}^{H_{0},\ldots ,H_{n}}(V_{1},\ldots ,V_{j},aV_{j+1},\ldots ,V_{n})-T_{f^{[n]}}^{H_{0},\ldots ,H_{n}}(V_{1},\ldots ,V_{j}a,V_{j+1},\ldots ,V_{n})=T_{f^{[n+1]}}^{H_{0},\ldots ,H_{n}}(V_{1},\ldots ,V_{j},[H_{j},a],V_{j+1},\ldots ,V_{n})}$

In combination with the operator trace ${\displaystyle {\rm {Tr}}}$ (or any other tracial function) the multiple operator integral satisfies the following cyclicity property:

${\displaystyle {\rm {Tr}}(V_{0}T_{f^{[n]}}^{H_{0},\ldots ,H_{n}}(V_{1},\ldots ,V_{n}))={\rm {Tr}}(V_{1}T_{f^{[n]}}^{H_{1},\ldots ,H_{n},H_{0}}(V_{2},\ldots ,V_{n},V_{0}))}$

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

The MOI admits several useful bounds. In terms of the operator norm, it follows that

${\displaystyle \|T_{\phi }^{H_{0},\ldots ,H_{n}}(V_{1},\ldots ,V_{n})\|\leq c_{\phi }\|V_{1}\|\cdots \|V_{n}\|,}$

for a constant ${\displaystyle c_{\phi }}$ that is independent of ${\displaystyle H_{0},\ldots ,H_{n}}$. If ${\displaystyle \phi }$ admits an explicit integral expression, an expression for ${\displaystyle c_{\phi }}$ can be thusly derived. In terms of Schatten norms ${\displaystyle \|\cdot \|_{p}}$, for ${\displaystyle p,p_{1},\ldots ,p_{n}\in [1,\infty ]}$ satisfying ${\displaystyle {\frac {1}{p}}={\frac {1}{p_{1}}}+\ldots +{\frac {1}{p_{n}}}}$ we have for the same constant ${\displaystyle c_{\phi }}$ that

${\displaystyle \|T_{\phi }^{H_{0},\ldots ,H_{n}}(V_{1},\ldots ,V_{n})\|_{p}\leq c_{\phi }\|V_{1}\|_{p_{1}}\cdots \|V_{n}\|_{p_{n}}.}$

If, moreover, ${\displaystyle p_{1},\ldots ,p_{n}\in (1,\infty )}$ (and consequently ${\displaystyle p\in (1,\infty )}$), and ${\displaystyle \phi =f^{[n]}}$, then

${\displaystyle \|T_{f^{[n]}}^{H_{0},\ldots ,H_{n}}(V_{1},\ldots ,V_{n})\|_{p}\leq c_{n}\|f^{(n)}\|_{\infty }\|V_{1}\|_{p_{1}}\cdots \|V_{n}\|_{p_{n}},}$

for a constant ${\displaystyle c_{n}}$ only dependent on ${\displaystyle n}$. The latter is a quite nontrivial but useful result,^[3] which in the double operator integral case ${\displaystyle n=1}$ implies that every Lipschitz function is actually operator-Lipschitz on ${\displaystyle {\mathcal {S}}^{p}}$, ${\displaystyle p\in (1,\infty )}$.^[4]

The terms in the Taylor expansion of ${\displaystyle f(H+V)}$ are multiple operator integrals. The Taylor expansion for ${\displaystyle \mathrm {Tr} (e^{z(H+V)})}$ coincides with the Dyson series.