In functional analysis, the multiple operator integral is a multilinear map informally written as

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

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A conceptually clean definition of the multiple operator integral is given as follows (it is a special case of both [1] and [2]). Let  , let   be a separable Hilbert space, and denote the space of bounded operators by  . Let   be possibly unbounded self-adjoint operators in  . For any function   (called the symbol) which admits a decomposition

 

for a certain finite measure space   and bounded measurable functions  , the multiple operator integral is the  -multilinear operator

 

defined by

 

for all  . One can show that the integrand is Bochner integrable, and (using Banach-Steinhaus) that   is a bounded multilinear map. Moreover,   only depends on   and   through  , as the notation   suggests.

Other definitions

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One may similarly define   on the product of Schatten classes   and end up with a mapping

 

where  . The restriction of the domain allows the multiple operator integral to be defined for a larger class of symbols  . Because one can (and often needs to) trade of assumptions on  ,  , and  , 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

 

for a von Neumann algebra   admitting a semifinite trace  . One then additionally assumes that   are affiliated to  .

Divided differences

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The most often used symbol of a multiple operator integral is the divided difference   of an   times continuously differentiable function  , defined recursively as

 
 

In particular,  , and

 

The multiple operator integral   is known to exist in the case that   in a suitable Besov space, for example, when  , and the multiple operator integral   for H\"older conjugate   (as above), is known to exist when   with   bounded.

Properties

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Proofs of the following facts can be found in

Algebraic properties

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The double operator integral has the following properties:

  1.  
  2.  

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

 

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.  
  2.  

In combination with the operator trace   (or any other tracial function) the multiple operator integral satisfies the following cyclicity property:

 

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

In quantum theory

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

  1. ^ Peller
  2. ^ ACDS