On some functional calculus of closed operators in a Banach space

We develop a functional calculus of closed operators in a Banach space based on the class of functions in the form $1/g$, where $g$ belongs to the class $R[a,b]$ introduced by M. G. Krein. We prove continuity, stability, uniqueness, spectral mapping, and inverse operator theorems and describe some other properties of the considered calculus.