Norm estimates for resolvents of non-selfadjoint operators having Hilbert-Schmidt inverse ones

The paper is devoted to an invertible linear operator whose inverse is a Hilbert - Schmidt operator and imaginary Hermitian component is bounded. Numerous regular differential and integro-differential operators satisfy these conditions. A sharp norm estimate for the resolvent of the considered operator is established. It gives us estimates for the semigroup and so-called Hirsch operator functions. The operator logarithm and fractional powers are examples of Hirsch functions. In addition, we investigate spectrum perturbation and suggest the multiplicative representation for the resolvent of the considered operator.