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
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] 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
, then the double operator integral
can be defined as for bounded and measurable functions
However one can look at more general operators
Consider the case where
be two bounded self-adjoint operators on
be a function on a set
is the identity operator.
μ − λ
denote the corresponding spectral measures of