Resolvents of operators inverse to Schatten–von Neumann ones

The paper deals with linear operators in a Hilbert space, whose inverse ones belong to the Schatten–von Neumann ideal of compact operators, and whose imaginary Hermitian components are bounded. A sharp norm estimate for the resolvents of the considered operators is derived. That estimate enables us to investigate spectrum perturbations and to establish bounds for the norms of the semigroups and Hirsch operator functions. The operator logarithm and fractional powers are examples of the Hirsch functions.