In functional analysis, double operator integrals (DOI) are integrals of the form
where is a bounded linear operator between two separable Hilbert spaces,
are two spectral measures, where stands for the set of orthogonal projections over , and is a scalar-valued measurable function called the symbol of the DOI. The integrals are to be understood in the form of Stieltjes integrals.
Double operator integrals can be used to estimate the differences of two operators and have application in perturbation theory. The theory was mainly developed by Mikhail Shlyomovich Birman and Mikhail Zakharovich Solomyak in the late 1960s and 1970s, however they appeared earlier first in a paper by Daletskii and Krein.[1]
Double operator integrals
[edit]The map
is called a transformer. We simply write , when it's clear which spectral measures we are looking at.
Originally Birman and Solomyak considered a Hilbert–Schmidt operator and defined a spectral measure by
for measurable sets , then the double operator integral can be defined as
for bounded and measurable functions . However one can look at more general operators as long as stays bounded.
Examples
[edit]Perturbation theory
[edit]Consider the case where is a Hilbert space and let and be two bounded self-adjoint operators on . Let and be a function on a set , such that the spectra and are in . As usual, is the identity operator. Then by the spectral theorem and and , hence
where and denote the corresponding spectral measures of and .
Literature
[edit]- Birman, Mikhail Shlemovich; Solomyak, Mikhail Zakharovich (1967). "Double Stieltjes operator integrals". Topics of Math. Physics. 1. Consultants Bureau Plenum Publishing Corporation: 25–54.
- Birman, Mikhail Shlemovich; Solomyak, Mikhail Zakharovich (1968). "Double Stieltjes operator integrals. II". Topics of Math. Physics. 2. Consultants Bureau Plenum Publishing Corporation: 19–46.
- Peller, Vladimir V. (2016). "Multiple operator integrals in perturbation theory". Bull. Math. Sci. 6: 15–88. arXiv:1509.02803. doi:10.1007/s13373-015-0073-y. S2CID 119321589.
- Birman, Mikhail Shlemovich; Solomyak, Mikhail Zakharovich (2002). Lectures on Double Operator Integrals.
- Carey, Alan; Levitina, Galina (2022). "Double Operator Integrals". Index Theory Beyond the Fredholm Case. Lecture Notes in Mathematics. Lecture Notes in Mathematics. Vol. 232. Cham: Springer. pp. 15–40. doi:10.1007/978-3-031-19436-8_2. ISBN 978-3-031-19435-1.
References
[edit]- ^ Daletskii, Yuri. L.; Krein, Selim G. (1956). "Integration and differentiation of functions of Hermitian operators and application to the theory of perturbations". Trudy Sem. Po Funktsion. Analizu (in Russian). 1. Voronezh State University: 81–105.
- ^ Birman, Mikhail S.; Solomyak, Mikhail Z. (2003). "Double Operator Integrals in a Hilbert Space". Integr. Equ. Oper. Theory. 47 (2): 136–137. doi:10.1007/s00020-003-1157-8. S2CID 122799850.
- ^ Birman, Mikhail S.; Solomyak, Mikhail Z. (2002). Lectures on Double Operator Integrals.