Double operator integral

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

 

and so[2][3]

 

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
  1. ^ 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.
  2. ^ 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.
  3. ^ Birman, Mikhail S.; Solomyak, Mikhail Z. (2002). Lectures on Double Operator Integrals.