Geodesic
Double operator integrals and their estimates in the uniform norm
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 24, Tome 232 (1996), pp. 148-173 
Yu. B. Farforovskaya. Double operator integrals and their estimates in the uniform norm. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 24, Tome 232 (1996), pp. 148-173. http://geodesic.mathdoc.fr/item/ZNSL_1996_232_a12/
@article{ZNSL_1996_232_a12,
     author = {Yu. B. Farforovskaya},
     title = {Double operator integrals and their estimates in the uniform norm},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {148--173},
     year = {1996},
     volume = {232},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1996_232_a12/}
}
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In the paper the conditions are considered for the existence of the double operator integral $\iint\varphi(\lambda,\mu)\,dE_\lambda TdF_\mu$, where $E_\lambda,F_\mu$ are the spectral functions of two self adjoint operators $A,B$ on a Hilbert space and $T$ is a bounded operator. In principal, the case where $A$ has finite spectrum is studied. Non-linear estimates of $\|f(A)T-Tf(B)\|$ in terms of the norm of $\|AT-TB\|$ for $f\in\operatorname{Lip}1$ are deduced. Also, a formula for the Fréchet derivative is presented. Bibl. 16 titles.