14 KAZUAKI TAIRA

1.2 Generation Theorems of Feller Semigroups

If {Tt}to is a Feller semigroup on K, we define its infinitesimal generator 21 by the

formula

(1.3) %u = \imTtU~U ,

v

' no t

provided that the limit (1.3) exists in the space Co(K). More precisely, the gener-

ator 21 is a linear operator from the space CQ(K) into itself defined as follows.

(1) The domain JD(21) of 21 is the set

£(») = {u € C0(K); the limit (1.3) exists} .

(2) QU* = lim

n o

£ * p - , u e D(»).

The next theorem is a version of the Hille-Yosida theorem adapted to the present

context (cf. [12, Theorem 9.3.1 and Corollary 9.3.2]):

Theorem 1.3. (i) Let {Tt}to be a Feller semigroup on K and 21 its infinitesimal

generator. Then we have the following:

(a) The domain D(21) is everywhere dense in the space CQ(K).

(b) For each a 0, the equation (al — 21)w = / has a unique solution u in

D(21) for any f € CQ(K). Hence, for each a 0, the Green operator (al — 21)"1 :

Co (A') — • Co(K) can be defined by the formula

ti = ( * / - a ) " 1 / , feC0(K).

(c) For each a 0, the operator (al — 21)_1 is non-negative on the space C0(K):

f e C0(K), / 0 onK = » (al - 2Q" 1 / 0 on K.

(d) For each a 0, the operator (al — 21)-1 is bounded on the space C0(K)

with norm

\\(aI-%)-'\\-.

a

(ii) Conversely if 21 is a linear operator from the space CQ(K) into itself satisfying

condition (a) and if there is a constant ao 0 such that, for all a ao conditions

(b) through (d) are satisfied, then 21 is the infinitesimal generator of some Feller

semigroup {Tt}to on K.

Corollary 1.4. Let K be a compact metric space and let 21 be the infinitesimal

generator of a Feller semigroup on K. Assume that the constant function 1 belongs

to the domain D(21) of 21 and that we have for some constant c

211 - c on K.

Then the operator 21' = 21+cl is the infinitesimal generator of some Feller semigroup

on K.

Although Theorem 1.3 tells us precisely when a linear operator 21 is the infinites-

imal generator of some Feller semigroup, it is usually difficult to verify conditions

(6) through (d). So we give useful criteria in terms of the maximum principle (cf.

[12, Theorem 9.3.3 and Corollary 9.3.4]):