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

350 KATO
THEOREM 5.3. Let the assumptions of Theorem 5.2 be satisjed, so
that W+ exists and is semicomplete. W, is complete if and only ;f the
following two conditions are satis$ed: (i) J is a (U, , +)-asymptotic
left inwerse to J’, and (ii) W; = W+( U, , US; J’) exists. When these
conditions are met, Wi is also complete and
w;w+ = PI, w+w; = Pz. (5.6)
Proof. Suppose (i) and (ii) are true. It follows from Theorem 5.2
applied to the triple U, , U, , J’ that W; is semicomplete. Further-
more, in view of W+ = P,W+ , (5.2) gives W;W+ = PI. This
implies that %( Wi) 1 PI.!& . But since the opposite inclusion is also
true, we have %( W;) = PI& . Thus W; is complete. Since there is a
complete symmetry between W+ and W; , we have also W+ W; = Pz
and W+ is complete.
Suppose conversely that W+ is complete. Then %( W+) is dense in
P&, and so (5.2) implies the existence of W; , with W; W+ = PI .
Similarly (5.3) implies that s-lim (Jy - 1) Uz(t) Pz = 0, that is,
(i) is true.
THEOREM 5.4. Let W+ = W+( U, , UI; J) exist and be complete.
Let J’ be a (U, , +)-asymptotic left inverse to J. Then J’ E B($& , &)
is (U, , +)-equivalent to J’ if and only if J’ is a (U, , +)-asymptotic
left inverse to J.
Proof. Since J’ is a (U, , +)-asymptotic left inverse to J, the
same is true of J’ if and only if (J’ - y) JUI(t) PI N 0, which is
equivalent to (J’ - J’) U%(t) W+ N 0 by (5.4). But the last relation
implies the (U, , +)-equivalence of J’ and J’ because W+W; = Pz
by Theorem 5.3.
6. PARTIALLY ISOMETRIC WAVE OPERATOW
For various reasons we are particularly interested in the case when
the wave operators are partially isometric.
THEOREM 6.1. Let W+ = W+( U, , VI; J) exist. In orde~ that W+
be a partial isometry with initial projection PI , it is necessary and suf-
$cient that