= X(s)x is weakly continuous. Equivalently, in view of Observation 1.2.2
above, a presentation of S on X is a norm-bounded weakly continuous function
on S to BW(X). It will be called
if X(t)X(s) = X(s)X(t) for s,t in
144- PROPOSITION. Let X be a presentation of S on X.
(i) The formula A{p)x
x G X defines a bounded operator A on
M(5) to B{X) with ||A|| =1 = sup{||A(s)|| : s e S}.
(ii) A weakly continuous operator T on X commutes with each A(p) if and
only if it commutes with each X(s).
(iii) If A is commuting then so is A, i.e., A(p)A(v) = A(v)A(p) for p,v in
(iv) If Z is an X*-closed convex subset (respectively cone, subspace) of X
invariant under X, i.e., under each X(s), then it is invariant under A(p)
for each p in MP(S) (respectively M
(5), M(S)). In particular, if X
fixes a point x of X then so does A restricted to M
(v) For each x G X, the map p —• A(p)x is weakly continuous on (M(5), rh)
(vi) For any norm-bounded subset £ of M(5) with A(£) C LW(X) (so that
A(£) C BW(X)), A : (£,rb) (^(XJjB^^J^-topoJogyJ is continuous.
(vii) Suppose that (X, X*) is a compatible pair. Then
(a) AM(S)CLW(X),
(b) A* defined by X*(s)p = pX(s) for s G S and p G l * is a presentation
of S on X* and £* = sup{||A*(s)|| : s G 5} ^ and
(cj if A* is the operator on M(S) to LW(X*) associated to A* then
A*(fi) = A(//)* for each fi in M(5).
Proof, (i), (iv) and (v) follow from observations made in §1.3 above, (vi) is
immediate from (i) and (v) and discussion in §1.2 above. For (ii) and (iii) we
first note if T is weakly continuous then for x G X, p G X*,
p(TA(n)x) = (T.p)(A(/i)x) - [(T.p)(\(8)x)dn(s) = f p((TX(s))x)dn(s).
Js Js
So if T commutes with X(s) then p(TA(p)x) = p(A(p)Tx), which gives (ii).
On the other hand, since X(t) is weakly continuous for each t, using the same
observation again,
p{A{v)A{p)x) = Jsp(X(t)A(p)x)du(t) = ^ ^p(X(t)(X(s)x)d»(s)^ du(t).
Similarly, p{A(p)A(v)x) = fs (fs(p(X(s)X(t)x)dv(t)) dp(s). But by Fubini's
Theorem this last integral is = fs (fs p(X(s)(X(t)x)d/j,(s)) dv(t). Hence if A is
commuting then A(/i)A(i/) = A(i/)A(/z), which gives (iii).
(vii) (a) Let p G M(5). The fact that A(p) G LW(X) essentially follows in
the same way as the second part of [3], Proposition 1.4, provided we make a
little modification there. As it stands, on [3], p. 220, line —9, i.e., line 9 from
below, we may apply the first part of the proof of [3], Proposition 1.2, in case
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