Double operator integrals and their estimates in the uniform norm

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.