Conditions for a linear closed operator in a Banach space are obtained that are necessary and sufficient to generate a resolving family of operators for a fractional differential equation. The cases of fractional derivatives of Riemann-Liouville, Hilfer, Dzhrbashyan Nersesyan are considered. These results are generalizations of the Hille-Yosida theorem for resolving semigroups of first-order equations. Similar theorems have been proved for analytic in a sector resolving families of operators. The results are used in the research of initial value problems for linear and quasilinear equations in Banach spaces. Abstract results are applied to the study of various initial boundary value problems for partial differential equations and their systems.