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.
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