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Multilinear Operator Integrals pp 65–112Cite as

Multiple Operator Integrals

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2250))

Abstract

Theory of multiple operator integrals arose as an extension of the double operator integration theory to the settings that could not be encompassed by the latter constructions. In particular, multilinear transformations naturally arise in finding summable approximations to operator functions in the case of nontrace class perturbations, as we will see in the next chapter. The first attempts to construct suitable multilinear extensions of double operator integrals were made in Solomjak and Sten’kin (Linear Operators and Operator Equations. Problems in Mathematical Analysis. Izdat Leningrad University, Leningrad, pp. 122–134, 1969), Pavlov (Linear Operators and Operator Equations. Problems in Mathematical Analysis. Izdat Leningrad University, Leningrad, pp. 99–122, 1969), Sten’kin (Izv Vysš Učebn Zaved Mat 179(4):102–115, 1977); the more recent approaches important for applications are due to Peller (J Funct Anal 233(2):515–544, 2006), Azamov et al. (Can J Math 61(2):241–263, 2009), Potapov et al. (Invent Math 193(3):501–538, 2013), Coine et al. (When do triple operator integrals take value in the trace class? arXiv:1706.01662). In this chapter we discuss the main constructions and properties of multiple operator integrals suitable for applications.

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Notes

  1. 1.

    Here the increment and decrement of the index i is understood modulo n, that is, if i = n, then i + 1 = 0 and if i = 0, then i − 1 = n.

  2. 2.

    If n is even, then a typical example is \(\begin {cases}k_i=i,& i\,{\leqslant }\,\frac {n}{2}\\ k_i=n-i+1,& i>\frac {n}{2}.\end {cases}\)

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, USA

    Anna Skripka

  2. School of Computer Science and Engineering, Inha University in Tashkent, Tashkent, Uzbekistan

    Anna Tomskova

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  1. Anna Skripka
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Skripka, A., Tomskova, A. (2019). Multiple Operator Integrals. In: Multilinear Operator Integrals. Lecture Notes in Mathematics, vol 2250. Springer, Cham. https://doi.org/10.1007/978-3-030-32406-3_4

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