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

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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. 
@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},
     publisher = {mathdoc},
     volume = {232},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1996_232_a12/}
}
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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/