A generation theorem for semigroups of growth order $\alpha $
Noboru Okazawa · Tohoku Mathematical Journal · 1974
Introduction.This paper is concerned with the generation of (operator) semigroups of growth order a.Extending the notion of a semigroup of class (C o ), Da Prato [1] introduces the notion of a semigroup of growth order n, n is a nonnegative integer.Roughly speaking, a semigroup {T(t); t > 0} of bounded linear operators on a Banach space is of growth order n if p Λ !Γ(ί)|| is bounded as t tends to zero; in particular, {T(t)} is of growth order 0 if and only if it belongs to class (C o ).In [1], Da Prato gave a characterization for the Laplace transform of t n T(t) through the notion of a closable linear operator of ftype n and its resolvent of order n.Namely, if A o is the infinitesimal generator of a semigroup {T(t)} of growth order n, then A o is of type n and its resolvent S(X, A o ) of order n is equal to the Laplace transform of t n T(t) and satisfies a certain stability condition.Viceversa if B is of type n and its resolvent S(λ, B) of order n satisfies the stability condition mentioned above, then there exists a unique semigroup of growth order n such that S(λ, B) = S(λ, A o ), where A o is the infinitesimal generator of the constructed semigroup.This result was generalized by Zafievskii [10] to the case of fractional a (cf.also Sobolevskii [8]).So, if it can be shown that B -A Q , then their result is proved to be a characterization for the infinitesimal generator of a semigroup of growth order a.But, this is not expected in general as noted in [2].The purpose of this paper is to give a characterization for the closure of the infinitesimal generator of a semigroup of growth order a.We first clarify some properties of the closure of the infinitesimal generator and then modify the construction of the semigroup stated in [1].In this way, we obtain a criterion for a closed linear operator in a Banach space to be the closure of the infinitesimal generator of a semigroup of growth order a.The main result of this paper is stated in § 1 and the proof of it is given in § 3 and §4.§2 is devoted to the preliminaries.