2 Integration of Unbounded Functions with Respect to Positive Operator Measures
Tải bản đầy đủ  0trang
5.2 Integration of Unbounded Functions with Respect …
105
We now consider a different, in general more restricted, approach to operator
integrals. For projection valued measures, however, the approaches will in the next
section be shown to coincide. Let D( f, E), or just D( f ), denote the set of those
ϕ ∈ H for which  f 2 is Eϕ,ϕ integrable.
Proposition 5.5 The set D( f, E) is a vector subspace of H contained in D( f, E).
Moreover, if ϕ ∈ D( f, E), then
 f  dEψ,ϕ  ≤
ψ
E(Ω)
for all ψ ∈ H, and the map ψ →
√
whose norm is at most
E(Ω)
 f 2 dEϕ,ϕ ,
f dEψ,ϕ on H is a bounded linear functional
 f 2 dEϕ,ϕ .
Proof Choose a sequence ( f n ) of simple functions converging pointwise to f , with
p
 f n  ≤  f  for all n ∈ N, and let f n = k=1 ck χ X k , with X 1 , . . . , X p ∈ A constituting
a partition of Ω. For each k = 1, . . . , p, let Yk1 , . . . , Ykrk ∈ A form a partition of
X k . Then for all ϕ, ψ ∈ H,
Eψ,ϕ (Yk jk ) =  ψ  E(Yk jk )ϕ  ≤ E(Yk jk )1/2 ψ
E(Yk jk )1/2 ϕ ,
which gives
p
p
rk
ck 
jk =1
k=1
p
ck 
p
E(Yk jk )1/2 ψ
ck 2 E(Yk jk )1/2 ϕ
k=1 jk =1
p
rk
rk
ψ  E(Yk jk )ψ
ck 2 ϕ  E(Yk jk )ϕ
k=1 jk =1
k=1 jk =1
p
p
ψ  E(X k )ψ
=
ck 2 ϕ  E(X k )ϕ
k=1
k=1
p
=
ψ  E(Ω)ψ
ck 2 ϕ  E(X k )ϕ
k=1
≤
E(Ω)
ψ
E(Yk jk )1/2 ϕ
rk
2
k=1 jk =1
=
E(Yk jk )1/2 ψ
jk =1
k=1
rk
≤
p
rk
Eψ,ϕ (Yk jk ) ≤
 f n 2 dEϕ,ϕ .
2
106
5 Operator Integrals and Spectral Representations: The Unbounded Case
Taking the suprema over all finite partitions of the sets X k one gets
 f n  dEψ,ϕ  ≤
ψ
E(Ω)
 f n 2 dEϕ,ϕ ,
where Eψ,ϕ  denotes the total variation of Eψ,ϕ . If ϕ ∈ D( f ), using the dominated
convergence theorem one obtains limn→∞  f n 2 dEϕ,ϕ =  f 2 dEϕ,ϕ . Hence, by
Fatou’s lemma,  f  is Eψ,ϕ integrable, and thus Eψ,ϕ integrable, showing that ϕ ∈
D( f ). This also proves the inequality stated in the proposition, and the last claim is
proved by observing that

f dEψ,ϕ  ≤
 f  dEψ,ϕ  ≤ lim
n→∞
≤
E(Ω)
ψ lim
=
E(Ω)
ψ
n→∞
 f n dEψ,ϕ 
 f n 2 dEϕ,ϕ
 f 2 dEϕ,ϕ .
Let now ϕ, ψ ∈ H, c, d ∈ C, and X ∈ A. Denoting ξ = cϕ + dψ we have
ξ  E(X )ξ = E(X )1/2 ξ
2
≤ c E(X )1/2 ϕ + d E(X )1/2 ψ
2
≤ 2c2 ϕ  E(X )ϕ + 2d2 ψ  E(X )ψ ,
which implies that D( f ) is a linear subspace.
Remark 5.1 If ϕ ∈ D( f ), then the Fréchet–Riesz theorem in conjunction with the
above proposition may be used to give a quick proof of the existence of what we have
denoted by L( f, E)ϕ, independently of the uniform boundedness principle used in
the proof of Lemma 5.1.
Remark 5.2 In general, the inclusion D( f ) ⊂ D( f ) in Proposition 5.5 may be strict.
For example, let μ : A → [0, 1] be a probability measure, and consider the positive operator measure X → E(X ) = μ(X )I . Take any measurable f which is
μintegrable and such that  f 2 is not μintegrable. Then {0} = D( f ) = D( f ) = H.
Choosing an f which is not μintegrable one gets the extreme case of D( f ) being
the null space.
Proposition 5.6 (a) If f is real valued, then L( f, E) is symmetric, that is, for any
ϕ, ψ ∈ D( f, E), ψ  L( f, E)ϕ = L( f, E)ψ  ϕ .
(b) If D( f, E) is a dense subspace of H, then the adjoint L( f, E)∗ is an extension
of L( f , E).
Proof (a) Let ( f n ) be a sequence of real simple functions converging pointwise to
f and satisfying  f n  ≤  f . By the dominated convergence theorem
5.2 Integration of Unbounded Functions with Respect …
107
ψ  L( f )ϕ = lim ψ  L( f n )ϕ = lim L( f n )∗ ψ  ϕ
n→∞
n→∞
L( f n )ψ  ϕ = L( f )ψ  ϕ ,
= lim
n→∞
where we have used the obvious fact that for a (simple) bounded real function f n the
operator L( f n ) is selfadjoint.
(b) Since f is measurable, f is also measurable, and the domains of the operators
L( f ) and L( f ) are the same, D( f ) = D( f ). If D( f ) is dense in H, then the adjoint
of L( f ) is defined. For each n ∈ N, let gn (x) = f (x) if  f (x) ≤ n, and gn (x) = 0
otherwise. Since gn is bounded, L(gn )∗ = L(gn ). By the dominated convergence
theorem we may write for all ψ, ϕ ∈ D( f ) = D( f )
ψ  L( f )ϕ = lim ψ  L(gn )ϕ
n→∞
= lim L(g n )ψ  ϕ = L( f )ψ  ϕ
n→∞
which shows that ψ
L( f ) ⊂ L( f )∗ .
∈
D(L( f )∗ ) and L( f )ψ
=
L( f )∗ ψ, that is,
5.3 Integration of Unbounded Functions with Respect
to Projection Valued Measures
Throughout this section, A ⊂ 2Ω is a σalgebra and E : A → L(H) is assumed to
be a projection valued measure. We first show that in this case the two approaches
to integration described in the previous section are actually equivalent.
Proposition 5.7 In the case of a projection valued measure E, we have D( f, E) =
D( f, E) for any measurable function f , and this subspace is dense in H.
Proof We already know that D( f, E) ⊂ D( f, E) (see Proposition 5.5). Now suppose
that ϕ ∈ D( f ). Choose a sequence ( f n ) of simple functions converging pointwise to
p
f , with  f n  ≤  f  for all n ∈ N. For a fixed n ∈ N we write f n = k=1 ck χ X k , where
X 1 , . . . , X p ∈ A are disjoint sets with union Ω. Then a simple calculation based on
the fact that the projections E(X j ) are pairwise orthogonal (by Proposition 4.6 (c))
shows that
p
 f n 2 dEϕ,ϕ =
2
ck E(X k )ϕ
2
=
f n dE(·)ϕ
k=1
with the obvious definition of the integral f n dE(·)ϕ of the simple function f n with
respect to the additive vector valued set function X → E(X )ϕ on A.
For any ψ the dominated convergence theorem shows that the sequence of the
numbers
108
5 Operator Integrals and Spectral Representations: The Unbounded Case
ψ
f n dE(·)ϕ =
f n dEψ,ϕ
tends to the limit f dEψ,ϕ as n → ∞. It thus follows from the uniform boundedness
principle that the sequence of the integrals  f n 2 dEϕ,ϕ is bounded, and so Fatou’s
lemma implies that  f 2 is integrable with respect to Eϕ,ϕ .
We now show that D( f ) is dense in H. Take some ϕ ∈ H. Write ϕ = E(Ω)ϕ + ψ
where ψ ⊥ E(Ω)(H). Since E(Ω)ψ = 0, it is clear that ψ ∈ D( f ), so that it is
enough to show that E(Ω)ϕ can be approximated by vectors from D( f ). For each
n ∈ N denote An = ω ∈ Ω  f (ω) ≤ n and ϕn = E(An )ϕ. As (An ) is an
increasing sequence, E(Ω)ϕ = limn→∞ ϕn (see Remark 4.4(b)). But ϕn ∈ D( f ),
since Eϕn ,ϕn (Ω\An ) = E(An )ϕ  (E(Ω) − E(An ))E(An )ϕ = 0, and f is bounded
on An .
In the sequel we use the notation D f for the space D( f, E) = D( f, E) for any
measurable function f : Ω → C.
Lemma 5.2 If f : Ω → C is a measurable function, then
L( f )ϕ
2
=
Ω
 f 2 dEϕ,ϕ
for all ϕ ∈ D f .
m
αi χ Ai is a simple function where the sets Ai ∈ A are disjoint
Proof If h = i=1
and their union is Ω, then
m
L(h)ϕ
2
m
=
αi E(Ai )ϕ
i=1
m
m
αi E(Ai )ϕ =
i=1
αi 2 ϕ  E(Ai )ϕ =
=
i=1
αi 2 E(Ai )ϕ
2
i=1
Ω
h2 dEϕ,ϕ .
The claim follows from this observation in the case when f is bounded, for we can
approximate such an f uniformly by simple functions (see Lemma 4.7) and then
apply Proposition 4.11. In the general case we define for all n ∈ N f n (ω) = f (ω) if
 f (ω) ≤ n, and f n (ω) = 0 if  f (ω) > n. Since f n is bounded, we have D f − fn =
D f , and using Proposition 5.5 and the dominated convergence theorem we find that
for all ϕ ∈ D f
L( f )ϕ − L( f n )ϕ = sup  ψ  L( f − f n )ϕ  ≤ sup
ψ ≤1
≤
Ω
 f − f n 2 dEϕ,ϕ
ψ ≤1 Ω
1
2
→0
 f − f n  dEψ,ϕ 
(5.1)