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]):
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