On the Characters and the Plancherel Formula of Nilpotent Groups ...

Condition 10.2.
SCATTERING THEORY WITfI TWO HILBERT SPACES 361
II UlP> - %k(t) /I& -+ 09
t-+&al.
It follows from (10.5) and (10.6) that (10.4) tends to zero as t --t & co,
showing that J is equivalent to J.
Let us now introduce an inverse identification operator
J’ E ~(~2 9 81) bY
Y’h 4 = e4 4, {u, 4 6 52 - (10.7)
J’ is simply the canonical injection of 5, into 5, . We shall show that
y is a (U, , &)-asymptotic left inverse to J. Since yf{u, o} = {Gu, V}
for {u, w> E $r , we have
II (1’9 - 1) Wt) ix, ~1 II = II (G - 1) W 11~1
with the notation used above. But since z++(t) E a(&), we have
/I (1 - G) ul(t> IIB, < 11 q(t) - %&) iiBl + 0
by Condition 10.2. Thus (fJ - 1) Ul(t) PI --+ 0, as we wished to
show.
Noting Theorems 5.3 and 9.3, we have thus proved
THEOREM 10.3. Suppose Conditions 9.1,10.1, and 10.2 are satisfied.
LetJ,JandJ’b e d$ e ne d as above. Then J is (U, , j-)-equivalent to J,
so that W, = W,( U, , UI; J) exist .and are equal to W,( U, , UI; J)
constructed in Theorem 9.3. In particular they are complete and partially
isometric with initial projection PI and $nal projection Pz . J’ is a
(U, , &)-asymptotic left inverse to J. Thus W,( U, , Uz; y) exist, are
complete and equal to W+ .
Remark 10.4. Theorem 10.3 is somewhat unsatisfactory in that
Condition 10.2 is not easy to verify directly. Thus it is desirable to
give some sufficient conditions for it.
THEOREM 10.5. Under Conditions 9.1 and 10.1, each of the follow-
ing conditions (all involving convergence in a) is suficient for Condition
10.2 to be satisfied:
(4
(b)
for each u E D(B,) n Q,R;
m= 1;
B,(edit~ - ebifBaC*) II 4 0, t+*=b