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

312 DUDLEY
6. SIMPLE SUBSETS OF HILBERT SPACE
In this section we study symmetric rectangular solids, ellipsoids,
and “octahedra” and determine when they are GC- and GB-sets. We
also study certain “full approximation sets” (see [14]), which are
maximal sets with a given adapted sequence {un}, while octahedra are
(among the) minimal sets.
For each class we shall have sequences {b,} of real numbers, b, JO,
related to an orthonormal set {vn> in H, usually complete. Let F,
be the subspace spanned by vi ,..., qn. For any orthonormal set
{qn} and any b, 2 0 we define the ellipsoid
Clearly, E is compact if and only if the b, for b, > 0 can be arranged
into a sequence b, JO. Then the {F~}, {F,) and {bn} are adapted to E.
(The F, are uniquely determined unless some positive b, are equal,
and the b, are unique.)
More abstractly, we can define a compact ellipsoid as an image
A(&) of the unit ball B, = {X : 11 x 11 < l> in H under a compact
operator A2
It follows that if E is a compact ellipsoid and S is a bounded linear
transformation from H into itself, then S(E) is a compact ellipsoid.
LEMMA 6.0. If E = E({b,), {c&) is a compact ellipsoid and P is
a f.d.p.,
P(E) = WQ, GM), bn 10, fin 109
then /3, < b, for all n.
Proof. We may assume the {q,} are complete. Given n let G, be
the linear span of #r ,..., #, . G, has at least one-dimensional intersection
with the set of vectors u orthogonal to P(yJ, j = l,..., n - 1.
If also u E P(E) then 11 u 11 < 11 Pv I/ for some v E E({b,}, {q~~}~&, so
11 u II < b, and hence p, < b, . Q.E.D.
Now we find the exponents of volume of ellipsoids.
PROPOSITION 6.1.
EV(E) = EW(E) = - ; - & .
2 For the equivalence of the definitions, see R. T. Prosser. The r-entropy and e-capa-
city of certain time-varying channels. J. Math. Anal. Appl. 16 (1966), 553-573.