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

Abstract

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