Local C-semigroups and local integrated semigroups.

*(English)*Zbl 0703.47031This paper is a contribution to the theory for C-semigroups (or R- semigroups) which are neither exponentially bounded nor defined on \([0,\infty)\) (local C-semigroups). The motivation of our work is found in the paper by E. B. Davies and M. M. H. Pang [Proc. London Math. Soc. 55, 181-208 (1987; Zbl 0651.47026)]:

(1) They pointed out that there exists a C-semigroup which is not exponentially bounded.

(2) They gave a number of examples to show that their theory is not contained in the theory of regular distribution semigroups, but they did not clarify the relationship between these two theories.

The generation problem for local C-semigroups is fairly general to the effect that the Laplace transform is not available. To overcome such difficulties, we introduce the notion of asymptotic C-resolvents close to that of C-resolvents by modifying the notion of asymptotic resolvents which was introduced to simplify the proof of the Kōmura generation theorem [T. Kōmura, J. Funct. Anal. 2, 258-296 (1968; Zbl 0172.407)] for locally equicontinuous semigroups on locally convex linear topological spaces. Making use of this notion we obtain a characterization of the complete infinitesimal generator of a local C- semigroup.

As an important subclass of local C-semigroups, we introduce the notion of local n-times integrated semigroups. The complete infinitesimal generator of a local n-times integrated semigroup has a non-empty resolvent set. This makes the characterization fairly simpler than that of a general local C-semigroup. Eventually it turns out that regular distribution semigroups can be characterized in terms of local integrated semigroups.

(1) They pointed out that there exists a C-semigroup which is not exponentially bounded.

(2) They gave a number of examples to show that their theory is not contained in the theory of regular distribution semigroups, but they did not clarify the relationship between these two theories.

The generation problem for local C-semigroups is fairly general to the effect that the Laplace transform is not available. To overcome such difficulties, we introduce the notion of asymptotic C-resolvents close to that of C-resolvents by modifying the notion of asymptotic resolvents which was introduced to simplify the proof of the Kōmura generation theorem [T. Kōmura, J. Funct. Anal. 2, 258-296 (1968; Zbl 0172.407)] for locally equicontinuous semigroups on locally convex linear topological spaces. Making use of this notion we obtain a characterization of the complete infinitesimal generator of a local C- semigroup.

As an important subclass of local C-semigroups, we introduce the notion of local n-times integrated semigroups. The complete infinitesimal generator of a local n-times integrated semigroup has a non-empty resolvent set. This makes the characterization fairly simpler than that of a general local C-semigroup. Eventually it turns out that regular distribution semigroups can be characterized in terms of local integrated semigroups.

Reviewer: N.Tanaka

##### MSC:

47D06 | One-parameter semigroups and linear evolution equations |