Some classes of functions of a linear closed operator

A linear closed densely defined operator and some domain $\Omega$ lying in the regular set of the operator and containing the negative real semiaxis of the real line are specified in a Banach space. We assume that power estimates for the norm of the resolvent operator are known at zero and infinity. We use the Cauchy integral formula to introduce operator functions generated by scalar functions that are analytic in a certain domain not containing the origin and containing the complement of $\Omega$ and have power estimates for their absolute values at zero and infinity. We study some properties of operator functions, which were studied by the authors earlier for the case of an operator whose inverse operator is bounded; in particular, we study the multiplicative property.