Characterization of polynomially bounded α-resolvent families

The theory of α- resolvent families was developed as a generalization of the classical theory of strongly continuous semigroups and cosine operator families, to study the fractional evolution equations with Caputo derivative of order α. An important problem in semigroup theory and also for cosine operator families is to discuss different type of boundedness (in terms of their generator) for these families. In this paper, we study polynomially bounded α- resolvent families. We impose conditions on the resolvent of a closed and densely defined linear operator to be the generator of an α-resolvent family. We also show that these conditions are necessary in the case of Hilbert spaces. This generalizes the result by T. Eisner on polynomially bounded semigroups. Moreover, since α- resolvent families describe the solutions to fractional evolution equations, with this generation result, we discuss the existence and stability of solutions to these problems at the same time.