FELLER SEMIGROUPS WITH BOUNDARY CONDITIONS 15

Theorem 1.5. Let K be a compact metric space. Then we have the following

assertions:

(i) Let B be a linear operator from the space C(K) = CQ(K) into itself, and

assume that:

(a) The domain D(B) of B is everywhere dense in the space C(K).

(f3) There exists an open and dense subset KQ of K such that if u € D(B) takes

a positive maximum at a point XQ of KQ, then we have

Bu(x0) 0.

Then the operator B is closable in the space C(K).

(ii) Let B be as in part (i), and further assume that:

(/?') Ifu£ D(B) takes a positive maximum at a point x' of K, then we have

Bu(x') 0.

(7) For some c*o 0, the range R(ctoI — B) of ctol — B is everywhere dense in

the space C(K).

Then the minimal closed extension B of B is the infinitesimal generator of some

Feller semigroup on K.

Corollary 1.6. Let % be the infinitesimal generator of a Feller semigroup on a

compact metric space K and M a bounded linear operator on the space C(K) into

itself Assume that either M or 2t' = 51 + M satisfies condition (/?'). Then the

operator 21/ is the infinitesimal generator of some Feller semigroup on K.