Generation of strongly continuous resolving families of operators for equations with a distributed derivative

Conditions for a linear closed operator are obtained in terms of the location of its resolvent set and estimates for its resolvent and its derivatives, which are necessary and sufficient to generate a strongly continuous resolving family of operators by this operator. Some properties of such resolving families are proved, and a theorem on the unique solvability of the Cauchy problem for the corresponding linear inhomogeneous equation is obtained. The results are used to prove the unique solvability of initial boundary value problems for equations with polynomials of a self-adjoint elliptic differential operator with respect to spatial variables and with a distributed derivative in time.