Operator integration,perturbations, and commutators

Under mild assumption, integral representations of the form (*) $$f(A_1 ) \cdot \mathfrak{J} - \mathfrak{J} \cdot f(A_1 ) = \int {\int {\frac{{f(\mu ) - f(\lambda )}}{{\mu - \lambda }}} } dE_1 (\mu )(A_1 \mathfrak{J} - \mathfrak{J}A_0 )dE_0 (\mu ),$$ are justified. Here A_k, k=0, 1, is a self-adjoint operator in a Hilbert space H_k, is an operator from H₀ H₁; in general, all the operators are unbounded; E_k is the spectral measure of the operator A_k. On the basis of the representation (^*), estimates of the s-numbers of the operator \(f(A_1 ) \cdot \mathfrak{J} - \mathfrak{J} \cdot f(A_0 )\) in terms of the s-numbers of the operator \(A_1 \mathfrak{J} - \mathfrak{J}A_0\) are given. Analogous results are obtained for commutators and antocommutators.