On functions of a positive operator

For functions $\varphi(1/z)=b+\int_0^\infty(z+s)^{-1}d\sigma(s)$, $b>0$, $\sigma(s)$ nondecreasing, and $A$ a positive operator in a Banach space, the operators $\varphi(A^{-1})$ and $\varphi(A)$ are constructed, their products and superpositions are investigated and the moment inequality as well as other estimates are proved. The results are generalized to the case when $A^{-1}$ does not exist or is not bounded.
Bibliography: 21 titles.