12 UNBOUNDED NORMAL OPERATORS 413
•/, then g(x) ]> 0 almost everywhere. (Observe that
AU«+(jt) ^ U
almost everywhere in the set of x such that g* (x) > 0.)
Deduce that, for all u e H, we have \u\ e H and |(H)~|g|«|. (If FA(P) =
(A(I-f-AG)-1 -#|z;)for v e&i, show that FA(M~)^ |w|X and use (c) above and
Problem 24.)
(e) Generalize the results of (d) to the case where X is locally compact. (Let (Kn)
be a sequence of compact subsets which cover X, and such that each is contained in
the interior of the next. For each n, consider the space $fn of restrictions to Kn of
functions of the form U7, where /e ^J(X, /x,) and Supp(/) <=- Kn; apply (d) to each
of these spaces.)
12. UNBOUNDED NORMAL OPERATORS
(15.12.1) Let E be a separable Hilbert space, and let / denote the identity
mapping of E. A (not necessarily continuous) linear mapping Tof a subspace
dom(T) of E (the "domain" of T, which need not be closed) into E will be
called, by abuse of language, a not necessarily bounded operator on E, or
simply an unbounded operator on E. The graph T(T) (1.4) is a vector subspace
of E x E, and Tis said to be a closed operator if T(T) is closed in the product
space E x E. The kernel Ker(T) of a closed operator is closed in E, because it
may be identified with the intersection of T(T) and E x {0} in E x E.
Throughout, it is to be understood that an equality 7^ = T2 between two
unbounded operators on E implies the equality
dom^) == dom(r2).
(15.12.2) Let T be an unbounded operator on a Hilbert space E. Then, of the
three properties:
(i) dom(T) is closed in E; (ii) T is closed', (iii) T is continuous', any two
imply the third.
If T is continuous, then F(T) is closed in dom(T) x E and therefore also
in E x E if dom(jT) is closed in E. Next, if T is continuous on dom(T), then
it extends by continuity to a linear operator T which is continuous on
dom(TO = dom(T) (5.5.4), and T(T') is the closure of T(T) in E x E. Hence
if Tis closed we have r(T) = T(T), so that dom(T) is closed in E. Finally, if
dom(T) is closed in E and if F(T) is closed in E x E, then T is continuous by
the closed graph theorem (12.16.11).
(15.12.3) In what follows we shall be concerned with unbounded operators
T such that dom(T) is dense in E. Let F be the set of all yeE such that the
iiiiT '
.
H. PENi'SYUMI