Double operator integrals and their estimates in the uniform norm

In this paper, conditions are considered for the existence of the double operator integral ∫∫ φ(λ,μ)dEλTdFμ, where Eλ, Fμ 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. Nonlinear estimates of ||f(A)T-Tf(B)|| in terms of the norm of ||AT-TB|| for f ∈ Lip 1 are deduced. Also, a formula for the Fréchet derivative is presented. Bibliography: 16 titles. © 1998 Plenum Publishing Corporation.