Triangular representations of functions of operators with Schatten–von Neumann Hermitian components

Let (Formula presented.) be a separable Hilbert space with the unit operator I, let A be a bounded linear operator in (Formula presented.) with a Schatten–von Neumann Hermitian component (Formula presented.) ((Formula presented.) means the operator adjoint to A) and let (Formula presented.) be a function analytic on the spectra of A and (Formula presented.). For (Formula presented.) we obtain the representation in the form of the sum of a normal operator and a quasi-nilpotent Schatten–von Neumann operator (Formula presented.), and estimate the norm of (Formula presented.). That estimate gives us an inequality for the norm of the resolvent (Formula presented.) of (Formula presented.) (Formula presented.). Applications of the obtained estimate for (Formula presented.) to operator equations, whose coefficients are operator functions, and to perturbations of spectra are also discussed.