4 AJIT IQBAL SINGH

Xx(s)

= 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

COMMUTING

if X(t)X(s) = X(s)X(t) for s,t in

S.

144- PROPOSITION. Let X be a presentation of S on X.

(i) The formula A{p)x —

Xx{p),

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

U(S).

(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

p

(5).

(v) For each x G X, the map p —• A(p)x is weakly continuous on (M(5), rh)

toX.

(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