7 Linear Operators on Hilbert Tensor Products and the Partial Trace
Tải bản đầy đủ - 0trang
56
3 Classes of Compact Operators
Lemma 3.4 shows that whenever
n
n
j=1
m
j=1
⊗ ηk , then
n
Sξk ⊗ T ηk ≤ S
Sϕ j ⊗ T ψ j −
m
k=1 ξk
ϕj ⊗ ψj =
k=1
m
ξk ⊗ ηk = 0.
ϕj ⊗ ψj −
T
j=1
k=1
Thus the map nj=1 ϕ j ⊗ ψ j → nj=1 Sϕ j ⊗ T ψ j is well defined on a dense subspace of H ⊗ K and since it is clearly linear, we may extend it to a bounded linear
map on the whole of H ⊗ K. Let us formulate this result in a proposition.
Proposition 3.1 Let H and K be Hilbert spaces and S ∈ L(H) and T ∈ L(K).
There is a unique operator S ⊗ T ∈ L(H ⊗ K) such that
(S ⊗ T )(ϕ ⊗ ψ) = Sϕ ⊗ T ψ
for all ϕ ∈ H and ψ ∈ K.
The linear operator S ⊗ T of the above proposition is called the (Hilbert) tensor
product of the operators S and T . Next we list some basic properties of the tensor
product operators.
Proposition 3.2 Let H and K be Hilbert spaces and S, S1 , S2 ∈ L(H) and
T, T1 , T2 ∈ L(K). Then
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
α(S ⊗ T ) = (αS) ⊗ T = S ⊗ (αT ) for all α ∈ C;
(S1 + S2 ) ⊗ T = S1 ⊗ T + S2 ⊗ T ;
(S1 ⊗ T1 )(S2 ⊗ T2 ) = S1 S2 ⊗ T1 T2 ;
(S ⊗ T )∗ = S ∗ ⊗ T ∗ ;
if S and T are selfadjoint, then also S ⊗ T is selfadjoint;
if S and T are unitary, then also S ⊗ T is unitary;
if S ∈ P(H) and T ∈ P(K), then S ⊗ T ∈ P(H ⊗ K);
if S ∈ T (H) and T ∈ T (K), then S ⊗ T ∈ T (H ⊗ K) and
tr[S ⊗ T ] = tr[S]tr[T ];
(i) if ϕ1 , ϕ2 ∈ H and ψ1 , ψ2 ∈ K, then
|ϕ1 ⊗ ψ1 ϕ2 ⊗ ψ2 | = |ϕ1 ϕ2 | ⊗ |ψ1 ψ2 |.
Proof (a), (b) For all ϕ ∈ H and ψ ∈ K we have
α(S ⊗ T ) (ϕ ⊗ ψ) = α(Sϕ ⊗ Sψ) = α Sϕ ⊗ T ψ = (αS) ⊗ T (ϕ ⊗ ψ)
so that α(S ⊗ T ) = (αS) ⊗ T since finite sums of simple tensors form a dense
subspace of H ⊗ K. In the same way one sees that α(S ⊗ T ) = S ⊗ (αT ) and (S1 +
S2 ) ⊗ T = S1 ⊗ T + S2 ⊗ T .
(c) Again, for all ϕ ∈ H and ψ ∈ K we have (S1 ⊗ T1 )(S2 ⊗ T2 )(ϕ ⊗ ψ) =
(S1 ⊗ T1 )(S2 ϕ ⊗ T2 ψ) = S1 S2 ϕ ⊗ T1 T2 ψ so that the argument is concluded as
above.
3.7 Linear Operators on Hilbert Tensor Products and the Partial Trace
57
(d) Let ϕ1 , ϕ2 ∈ H and ψ1 , ψ2 ∈ K. One has
ϕ1 ⊗ ψ1 | (S ⊗ T )ϕ2 ⊗ ψ2 = ϕ1 ⊗ ψ1 | Sϕ2 ⊗ T ψ2 = ϕ1 | Sϕ2
= S ∗ ϕ1 | ϕ2 T ∗ ψ 1 | ψ 2 = S ∗ ϕ1 ⊗ T ∗ ψ 1 | ϕ2 ⊗ ψ 2
ψ1 | T ψ2
= (S ∗ ⊗ T ∗ )(ϕ1 ⊗ ψ1 ) | ϕ2 ⊗ ψ2 .
Since the linear combinations of the simple tensors are dense in the Hilbert tensor
product the claim follows.
The claim in (e) is an immediate consequence of (d).
(f) Clearly IH⊗K = IH ⊗ IK . Suppose that S and T are unitary. Then (S ⊗
T )∗ (S ⊗ T ) = (S ∗ ⊗ T ∗ )(S ⊗ T ) = S ∗ S ⊗ T ∗ T = IH ⊗ IK = (S ⊗ T )(S ⊗ T )∗
implying the claim.
(g) (S ⊗ T )2 = S 2 ⊗ T 2 = S ⊗ T = S ∗ ⊗ T ∗ = (S ⊗ T )∗ whenever S = S ∗ =
2
S and T = T ∗ = T 2 .
(h) Noting that |S ⊗ T |2 = (S ⊗ T )∗ (S ⊗ T ) = S ∗ S ⊗ T ∗ T = |S|2 ⊗ |T |2 =
(|S| ⊗ |T |)2 , we have |S ⊗ T | = |S| ⊗ |T | by the uniqueness of the positive square
and
root. Let K ⊂ H and L ⊂ K be orthonormal bases. As
ξ∈K ξ | |S|ξ
η∈L η | |T |η < ∞, we have
ξ ⊗ η | |S ⊗ T |ξ ⊗ η =
ξ∈K , η∈L
ξ | |S|ξ
η | |T |η
ξ∈K , η∈L
ξ | |S|ξ
=
ξ∈K
η | |T |η < ∞,
η∈L
so that S ⊗ T ∈ T (H ⊗ K). Since the sets ( ξ | Sξ )ξ∈K and ( η | T η )η∈L are summable, we get
ξ ⊗ η | (S ⊗ T )(ξ ⊗ η) =
tr[S ⊗ T ] =
ξ∈K , η∈L
ξ | Sξ
=
ξ∈L
ξ | Sξ
η |Tη
ξ∈K , η∈L
η | T η = tr[S]tr[T ].
η∈L
(i) It suffices to observe that either side applied to ξ ⊗ η for any ξ ∈ H and η ∈ K
yields ϕ2 |ξ ψ2 |η (ϕ1 ⊗ ψ1 ).
Remark 3.4 All the above properties of tensor product operators S ⊗ T can be generalised in a straightforward way to the case of general tensor product operators
S1 ⊗ · · · ⊗ Sn on Hilbert tensor products H1 ⊗ · · · ⊗ Hn . We may also use obvious
associativity properties without explicit justification.
We conclude this section by examining the trace class operators of the Hilbert
tensor product H ⊗ K. We see that any operator T ∈ T (H ⊗ K) defines two trace
class operators TI ∈ T (H) and TI I ∈ T (K) through a linear map called the partial
trace.
58
3 Classes of Compact Operators
Proposition 3.3 If T ∈ T (H ⊗ K), there is a unique TI ∈ T (H) such that
tr[TI A] = tr[T (A ⊗ IK )]
(3.13)
+
for all A ∈ L(H). In particular, if T ∈ Ts (H ⊗ K)+
1 , then TI ∈ Ts (H)1 .
Proof First we show that if there is a trace class operator TI satisfying the condition (3.13), it is unique. Suppose that TI is another such operator. The substitution A = |ψ ϕ| in (3.13) yields ϕ | TI ψ = ϕ | TI ψ . Since this is valid for any
ϕ ∈ H, we have TI = TI . Next we show that such an operator exists. According to
Remark 3.1 we may write T = ∞
n=1 cn |ψn ϕn | where cn ≥ 0 and (ψn ) and (ϕn )
are orthonormal sequences in H ⊗ K. If for each n there is an operator Sn ∈ T (H)
such that tr[Sn A] = tr[|ψn ϕn |(A ⊗ IK )] for all A ∈ L(H), then we can take TI =
∞
n=1 cn Sn . We may thus assume that T = |ψ ϕ| for some ϕ, ψ ∈ H ⊗ K. Choose
orthonormal bases K ⊂ H, L ⊂ K. Denoting ϕη = ξ∈K ξ ⊗ η | ϕ ξ and ψη =
ξ∈K ξ ⊗ η | ψ ξ for all η ∈ L, one immediately sees that ϕ =
η∈L ϕη ⊗ η, ψ =
2
2
2
2
= ϕ and η∈L ψη = ψ . Define the operator
η∈L ψη ⊗ η,
η∈L ϕη
TI = η∈L |ψη ϕη |. This operator belongs to the trace class since |ψη ϕη | 1 =
ψη ϕη and
ψη
ϕη ≤
η∈L
ψη
η∈L
ϕη
2
2
= ϕ ψ < ∞.
η∈L
We now have
ϕη1 ⊗ η1 | Aψη2 ⊗ η2
tr |ψ ϕ|(A ⊗ IK ) = ϕ | (A ⊗ IK )ψ =
η1 , η2 ∈L
=
ϕη | Aψη = tr[TI A].
η∈L
Thus TI satisfies the condition (3.13). Suppose now that T is positive and choose
any ϕ I ∈ H. One finds that
ϕ I | TI ϕ I = tr T (|ϕ I ϕ I | ⊗ IK ) =
ϕI ⊗ η | T ϕI ⊗ η ≥ 0
η∈L
implying that TI ≥ 0. Substituting A = IH in (3.13) one sees that tr[T ] = tr[TI ] and
thus especially when T is of trace 1, then TI is also of trace 1.
Remark 3.5 The map T → TI from T (H ⊗ K) to T (H) defined through Eq. (3.13)
is easily seen to be linear. Proposition 3.3 also shows that this map is trace preserving (i.e. tr[TI ] = tr[T ]) and positive (i.e. TI ≥ 0 if T ≥ 0). The map T → TI
of Proposition 3.3 is called the partial trace (over the Hilbert space K) and it is
denoted TI = tr K [T ] = tr I I [T ]. It is obvious that we may define the partial trace
tr H : T (H ⊗ K) → T (K) over H in a completely analogous manner.
3.8 The Schmidt Decomposition of an Element of H1 ⊗ H2
59
3.8 The Schmidt Decomposition of an Element of H1 ⊗ H2
The formula in the next theorem gives the Schmidt decomposition (also known as
the polar or biorthogonal decomposition) of an element in the Hilbert tensor product
of two Hilbert spaces. Its proof involves in a crucial way some of the most central
techniques developed above, like the polar decomposition of an operator and the the
spectral theory of compact operators. In a sense it wraps up the theory we have seen
so far. We let H1 and H2 denote two Hilbert spaces.
Theorem 3.15 (a) Any nonzero Ψ ∈ H1 ⊗ H2 can be expressed as the sum of a
norm convergent series
λi ϕi ⊗ ψi
Ψ =
i
where (ϕi ) resp. (ψi ) is an orthonormal sequence (finite or infinite) in H1 resp. H2 ,
and each λi > 0. In this kind of representation we always have i λi 2 = Ψ 2 .
(b) Suppose that
μi ξi ⊗ ηi
Ψ =
i
is another representation having the properties mentioned in (a). Assume that
λ1 ≥ λ2 ≥ · · · and μ1 ≥ μ2 ≥ · · · . Then λi = μi for each i. For any i the sets
{ϕ j | λ j = λi } and {ξ j | μ j = μi } span the same (finite-dimensional) subspace of
H1 , and similarly the sets {ψ j | λ j = λi } and {η j | μ j = μi } span the same subspace
of H2 . In case some number λi (and hence also μi ) occurs only once, we must have
ξi = ci ϕi and ηi = ci−1 ψi for some ci ∈ C with |ci | = 1.
Proof We first consider the special case where H1 = H2 = H. Choose an orthonormal basis K for H and let J : H → H be defined as the conjugate-linear isometric
map which changes the coefficients of any ϕ ∈ H in the expansion with respect
to K to their complex conjugates. Define the bilinear map f : H × H → HS(H)
via f (ϕ, ψ) = |ϕ J ψ|. Using Theorems 3.6, 3.3 (b), 3.11 and 2.20 we get, for any
ϕ, ψ, ξ, η ∈ H,
f (ϕ, ψ) | f (ξ, η) = tr f (ϕ, ψ)∗ f (ξ, η) = tr |J ψ ϕ||ξ J η|
= ϕ|ξ
Jη | Jψ = ϕ | ξ
ψ|η .
Since the finite rank operators are dense in HS(H), we are thus allowed to use
Lemma 2.4 to conclude that there is an isometric isomorphism g : H ⊗ H → HS(H)
satisfying g(ϕ ⊗ ψ) = f (ϕ, ψ). Using Remark 3.1 (b) we find some positive numbers λ1 ≥ λ2 ≥ · · · and some (finite or infinite) orthonormal sequences (ϕi ) and
(J ψi ) such that
g(Ψ ) =
λi |ϕi J ψi |
i
60
3 Classes of Compact Operators
(convergence in the operator norm and also in the Hilbert–Schmidt norm by
Theorem 3.11 (d)). As J = J −1 preserves orthogonality, (ψi ) is also an orthonormal sequence. We have Ψ = g −1 [ i λi |ϕi J ψi |] = i λi g −1 (|ϕi J ψi |) =
i λi ϕi ⊗ ψi , the last sum converging in the norm of H ⊗ H. We have thus proved
the first claim of (a) in this special case. The equation involving the norm also follows
for this particular construction. After we have proved (b), we see it generally (or else
prove it by a direct calculation, exercise).
We now consider the uniqueness part (b), still with H1 = H2 = H. Since g is an
isometry, we get
λi |ϕi J ψi | = g(Ψ ) = g
μi ξi ⊗ ηi
i
i
=
μi |ξi J ηi |.
i
From Remark 3.2 we see at once the equalities λi = μi . Moreover, the linear spans
lin{J ψ j | λ j = λi } and lin{J η j | μ j = μi } are the same, and so are lin{ϕ j | λ j = λi }
and lin{ξ j | λ j = λi }. Since J (lin{ψ j | λ j = λi }) = lin{J ψ j | λ j = λi } and similarly
we
have
lin{ψ j | λ j = λi } =
J (lin{η j | μ j = μi }) = lin{J η j | μ j = μi },
lin{η j | μ j = μi }. Suppose now that some λi occurs only once. In this case we get
ξi = αϕi and ηi = βψi for some α, β ∈ C. The maps |ϕi J ψi | and |ξi J ηi | take
the same value (= λi−1 g(Ψ )(J ψi )) at J ψi , and so J ψi |J ψi ϕi = J βψi |J ψi αϕi
implying 1 = βα and hence the last claim in (b), since all the vectors ϕi , ψi , ξi , ηi
have norm one.
In the general case we may clearly assume that H1 ⊂ H2 or H2 ⊂ H1 . As our
claims are invariant under the isometric isomorphism from H1 ⊗ H2 onto H2 ⊗ H1
mapping ϕ ⊗ ψ to ψ ⊗ ϕ, we may in fact assume that H1 is just a closed subspace of
H2 . Take H = H2 in the first part of the proof. Let D denote the closed linear span of
the set {ξ ⊗ η | ξ ∈ H1 , η ∈ H}. The map (ξ, η) → ξ ⊗ η from H1 × H to D clearly
satisfies the hypotheses of Lemma 2.4, and so it determines an isometric isomorphism
from H1 ⊗ H onto D. We identify H1 ⊗ H with D via this isomorphism. The whole
theorem will be proved if we show that whenever Ψ ∈ D with Ψ = i λi ϕi ⊗ ψi
as in the statement of (a), then necessarily every ϕi belongs to H1 . We observe
that the range of g(Ψ ) is contained in H1 for every Ψ ∈ D, since this is clearly
true for g(ϕ ⊗ ψ) = |ϕ J ψ| where ϕ ∈ H1 , and an arbitrary g(Ψ ) with Ψ ∈ D
can be approximated in (the Hilbert–Schmidt norm and hence also in) the operator
norm by finite sums of operators of the type g(ϕ ⊗ ψ). But in the representation
g(Ψ ) = i λi ϕi ⊗ ψi each ϕi = λi−1 g(Ψ )(J ψi ) is in the range of g(Ψ ).
We have the following useful application.
Proposition 3.4 Let H1 and H2 be Hilbert spaces and
Ψ =
λi ϕi ⊗ ψi
i
the Schmidt decomposition of a unit vector Ψ ∈ H1 ⊗ H2 . Then for the partial trace
operators of the rank one projection |Ψ Ψ | we have the formulas
3.8 The Schmidt Decomposition of an Element of H1 ⊗ H2
tr H2 |Ψ Ψ | =
61
λi2 |ϕi ϕi |
i
and
tr H1 |Ψ Ψ | =
λi2 |ψi ψi |.
i
Proof Denote Ψn =
n
i=1
λi ϕi ⊗ ψi . Then
n
n
|Ψn Ψn | =
n
n
λi λ j |ϕi ⊗ ψi ϕ j ⊗ ψ j | =
i=1 j=1
λi λ j |ϕi ϕ j | ⊗ |ψi ψ j |
i=1 j=1
by Proposition 3.2 (i). For any A ∈ L(H1 ) we get
tr (|ϕi ϕ j | ⊗ |ψi ψ j |)(A ⊗ IH2 ) = tr |ϕi ϕ j |A tr |ψi ψ j |
= tr |ϕi ϕ j |A ψ j |ψi
(see Proposition 3.2 (h) and Theorem 3.11 (b)), and since the vectors ψ j are orthonormal, we obtain
n
tr |Ψn Ψn |(A ⊗ IH2 ) =
n
λi2 tr
|ϕi ϕ j |A = tr
i=1
λi2 |ϕi ϕi | A .
i=1
Since the series i λi2 |ϕi ϕ j | converges in the trace norm (see Theorem 3.10 (b)
and Theorem 3.11 (d)) and |Ψn Ψn | → |Ψ Ψ | in the trace norm (as can be seen
using Theorem 3.11 (a)), from Theorem 3.11 (a) it follows that
tr |Ψ Ψ |(A ⊗ IH2 ) = tr
λi2 |ϕi ϕi | A .
i
This proves the claim for tr H2 [|Ψ Ψ |], and the proof for tr H1 [|Ψ Ψ |] is similar.
Remark 3.6 The above proposition can be used to give an alternative proof for Proposition 3.3, thus reducing the partial trace result to the Schmidt decomposition.
3.9 Exercises
1. Let g : N → C be a bounded function and Tg : 2 → 2 the bounded linear
operator defined by the formula Tg f = g f (see exercise 22 in Chap. 2). Show
that Tg is a compact operator if limn→∞ g(n) = 0.
2. Is the condition limn→∞ g(n) = 0 also necessary for the operator Tg in the
preceding exercise to be compact?
62
3 Classes of Compact Operators
3. Let T = V A be the polar decomposition of a compact operator T . Is
(a) the operator V ,
(b) the operator A
necessarily compact?
4. Does there exist any injective compact operator T ∈ L( 2 )?
5. Does there exist any surjective compact operator T ∈ L( 2 )?
6. Let T ∈ L(H) be a positive compact operator. Show that the square root of T is
compact. (Hint: use the spectral representation.)
7. Let I = {0} be a (not necessarily closed) two-sided ideal of L(H). Show that I
contains every finite rank operator.
8. Using the fact that a subset of a metric space is compact if and only if it is
precompact and complete, show that the space C(H) of the compact operators
on H is norm closed in L(H).
9. Let g : N → C be a bounded function and Tg : 2 → 2 the bounded linear map
defined via the formula Tg f = g f . Show that Tg is a Hilbert–Schmidt operator
if and only if g ∈ 2 .
10. Show that, in the situation of the preceding exercise, Tg belongs to the trace class
T ( 2 ) if and only if g ∈ 1 , i.e. ∞
n=1 |g(n)| < ∞.
11. Find some isometric isomorphism between the Hilbert space HS(H) (of the
Hilbert–Schmidt operators on H) and a suitable Hilbert sum ⊕
x∈X Hx where
always Hx = H.
12. Let I ⊂ L(H) be a vector subspace such that ST ∈ I whenever S ∈ I and
T ∈ L(H). (Thus I is a right ideal.) Show that I is a two-sided ideal if and only
if S ∗ ∈ I whenever S ∈ I.
13. Let S, T ∈ L(H) be compact operators. Show that the map X → S X T from the
Banach space L(H) to L(H) is a compact operator. (The definition of a compact
operator is the same as in the case of a Hilbert space.)
14. Let S, T ∈ L(H) \ {0} be such that the map X → S X T from the Banach space
L(H) to L(H) is a compact operator. Show that S and T are compact operators.
15. Prove the formula J η | J ψ = ψ | η which was used in the proof of Theorem 3.15.
16. Prove by a direct calculation the formula i λi 2 = Ψ 2 in Theorem 3.15.
17. Suppose that the Schmidt decomposition of Ψ ∈ H1 ⊗ H2 has only a finite
number p of terms. Is it possible that Ψ might be written as a sum of fewer
than p terms of the form ϕ ⊗ ψ with ϕ ∈ H1 , ψ ∈ H2 (with no requirement of
orthogonality)?
18. Let H be a finite-dimensional Hilbert space with an orthonormal basis
{ 1 , . . . , n }. Show that there is an inner product preserving isomorphism from H
onto Cn mapping each i to ei = (0, . . . , 0, 1, 0 . . . , 0). For simplicity, consider
the tensor product H ⊗ H. Show that H ⊗ H is isomorphic as a Hilbert space
to Mn (C), the Hilbert space of n × n matrices equipped with the inner product
A | B = tr A∗ B where tr C is the sum of the main diagonal elements of the
matrix C. Interpret and reprove the content of Theorem 3.15 in this situation by
considering the polar decomposition of a matrix. Does one avoid invoking the
map J used in the proof of Theorem 3.15 or is it lurking somewhere?
Chapter 4
Operator Integrals and Spectral
Representations: The Bounded Case
In the preceding chapter the spectral theory of compact selfadjoint operators played
a key role. One purpose of this chapter is to present the corresponding theory for
arbitrary bounded selfadjoint operators. We begin with auxiliary techniques from
the (scalar) theory of measure and integration. Then the notion of positive operator
measure is introduced and studied. This is a key concept for the rest of the book, but in
this chapter its special case, spectral measure, is needed for the spectral representation
theory alluded to above. We also consider the two-variable case of positive operator
bimeasures, an important notion for our physical applications, but at the same time a
tool for the spectral representation of bounded normal, especially unitary, operators.
The latter will have an instrumental role in the next chapter dealing with unbounded
selfadjoint operators.
4.1 Classes of Sets and Positive Measures
In this chapter we first collect some basic material on measure and integration needed
later. We omit many proofs. They can of course be found in many sources, but we
mention especially [1, 2] whose presentations are fairly close to ours.
Throughout this chapter, Ω is a set. We denote by 2Ω the set of its subsets. A
collection of subsets A ⊂ 2Ω is called a σ-algebra if ∅ ∈ A and Ac (=Ω\A) and
Ω
∪∞
n=1 An belong to A whenever A ∈ A and An ∈ A for all n ∈ N. If F ⊂ 2 , the
intersection of the σ-algebras containing F is clearly a σ-algebra; it is called the σalgebra generated by F. In particular, if Ω is a topological space (in practice usually
R or Rn ), the σ-algebra generated by its topology (i.e. the class of open sets) is called
the Borel σ-algebra of Ω and denoted by B(Ω). The sets B ∈ B(Ω) are Borel sets.
We may use without explicit mention the easily proved fact that if X is a subset of
a topological space Ω, then its Borel σ-algebra, i.e. the σ-algebra generated in 2X
by the relative (subspace) topology of X, is the same as the set of the intersections
© Springer International Publishing Switzerland 2016
P. Busch et al., Quantum Measurement, Theoretical and Mathematical Physics,
DOI 10.1007/978-3-319-43389-9_4
63
64
4 Operator Integrals and Spectral Representations: The Bounded Case
B ∩ X where B ∈ B(Ω). We may also call any set A ∈ A A-measurable. If A ⊂ 2Ω
is a σ-algebra, we call the pair (Ω, A) a measurable space.
We occasionally need the following somewhat more general notions:
Definition 4.1 Let Ω be a set.
(a) We say that R ⊂ 2Ω is a ring (of sets) if
(i) ∅ ∈ R;
(ii) E \ F ∈ R whenever E, F ∈ R;
(iii) E ∪ F ∈ R whenever E, F ∈ R.
(b) A ring R ⊂ 2Ω containing Ω is called an algebra (of sets).
(c) We say that S ⊂ 2Ω is a semiring if
(i) ∅ ∈ S;
(ii) E ∩ F ∈ S whenever E, F ∈ S;
(iii) whenever E, F ∈ S, E\F is the union of a finite number of disjoint sets
belonging to S.
Since E ∩ F = E \ (E\F), a ring is closed with respect to the intersection of two
(and by induction a finite number of) sets. In particular, a ring is a semiring. It is
straightforward to show that if S ⊂ 2Ω is a semiring, then the set of finite unions
of sets belonging to S is a ring. We denote this ring by U(S). Clearly U(S) is the
smallest ring containing S. For any subset R ⊂ 2Ω , the smallest ring containing R
exists; it is the intersection of all the rings containing R and called the ring generated
by R.
Remark 4.1 It is instructive to compare the structure (P(H), ≤, ⊥ , 0, I) of the projection lattice P(H) of a Hilbert space H (see Theorem 2.14 and Remark 2.3) with the
corresponding structure (A, ⊂, c , ∅, Ω) of a σ-algebra A of subsets of a (nonempty)
set Ω. Both are lattices, and the respective mappings ⊥ and c are orthocomplementations (in the sense of Remark 2.3). Let L denote either of these orthocomplemented
lattices, with, for instance, a ∧ b and a ∨ b denoting the greatest lower bound (infimum) and the least upper bound (supremum) with respect to the relevant order. We
say that the elements a, b, c ∈ L form a distributive triple if the equalities
a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c),
a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)
hold, together with the other four equalities obtained by cyclic permutation of a, b, c.
It is obvious that any triple X, Y , Z ∈ A is distributive. This means that A is a distributive lattice. Distributive orthocomplemented lattices are generally called Boolean
algebras. On the other hand, it is an easy exercise to show that in the projection
lattice P(H) a triple P, Q, R is distributive if and only if the projections are pairwise
commutative. P(H) is not a Boolean algebra, unless dim(H) = 1. By Theorem 2.14,
P(H) is complete (i.e. every subset has supremum and infimum) whereas A is usually
only σ-complete (i.e. every countable subset has supremum and infimum).
4.2 Measurable Functions
65
4.2 Measurable Functions
In this section we let Ω and Λ be sets and A ⊂ 2Ω , B ⊂ 2Λ σ-algebras. The measurability of a function relates to σ-algebras as continuity relates to topologies.
Definition 4.2 (a) We say that a map f : Ω → Λ is (A, B)-measurable if f −1 (B) ∈
A for all B ∈ B.
(b) If, in the above, Λ is Rn and B is its Borel σ-algebra B(Rn ) then we call a
(A, B)-measurable map simply A-measurable or just a measurable function if
A is clear from the context. A similar remark applies when we have the set R of
the extended real numbers in place of Rn . (We equip R with its natural topology
which makes it homeomorphic with the interval [−1, 1] ⊂ R.)
We list without proof some basic properties of measurable functions. For the
(A, B)-measurability of f : Ω → Λ it suffices that for some class F ⊂ 2Λ generating
B, f −1 (B) ∈ A whenever B ∈ F. As a consequence, the following holds.
Proposition 4.1 For a function f : Ω → R the following conditions are equivalent:
(a)
(b)
(c)
(d)
(e)
(f)
f is A-measurable;
f −1 ([a, ∞]) ∈ A for each a ∈ R;
f −1 ((a, ∞]) ∈ A for each a ∈ R;
f −1 ([−∞, a]) ∈ A for each a ∈ R;
f −1 ([−∞, a)) ∈ A for each a ∈ R;
f −1 (U) ∈ A for each open set U ⊂ R, f −1 (∞) ∈ A and f −1 (−∞) ∈ A.
In particular, a function f : Ω → R is measurable if and only if it is measurable
when regarded as a function into the extended real line R. The above proposition
also easily implies that lim sup fk and lim inf fk for a sequence (fk ) of measurable
functions are measurable. In particular, the limit of a pointwise convergent sequence
of real or extended real valued measurable functions is measurable. The same applies
to Rn -valued functions, since a function f = (f1 , . . . , fn ) : Ω → Rn is measurable
if and only if the real valued functions f1 , . . . , fn are measurable. The measurable
functions f : Ω → Rn form a vector space with respect to the pointwise operations.
If f : Ω → Rn is a measurable function and g is a continuous Rm -valued function
defined on its range, then the composite function g ◦ f is measurable.
4.3 Integration with Respect to a Positive Measure
We now assume that (Ω, A, μ) is a measure space, which means that A ⊂ 2Ω is a
σ-algebra and μ : A → [0, ∞] is a measure, i.e. a nonnegative R-valued set function
∞
which is σ-additive in the sense that μ(∅) = 0 and μ(∪∞
n=1 An ) =
n=1 μ(An ) for any
sequence (An ) of disjoint sets belonging to A. If here μ(Ω) = 1, μ is a probability
measure and (Ω, A, μ) is a probability space.
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4 Operator Integrals and Spectral Representations: The Bounded Case
We assume that G is a Banach space whose scalar field is R or C; we use the unified
notation K to stand for either one. We are going to define the Bochner integral of a
G-valued function. (We use this term since the definition following [1] is equivalent
to one originally given by Bochner.) Mostly in the sequel it would suffice to consider
the special case where G is the scalar field R or C (in which case we obtain the
Lebesgue integral), but we consider a general Banach space when this generality
does not present any extra difficulty.
A simple function f : Ω → G is a function which only takes a finite number of
values, and the inverse image of each of these values belongs to A. If, moreover, f
vanishes outside a set of finite measure, f is an integrable simple function, and the
integral of f is defined as
n
Ω
f dμ =
Ω
f (ω) dμ(ω) =
μ(Ei )xi ,
i=1
where f (Ω) = {x1 , . . . , xn } and Ei = f −1 (xi ). (We use the convention ∞ · 0 = 0
even if 0 is the zero element of a Banach space.) The integral of certain more general
functions is defined by approximating them in a way described below.
We say that the sequence of functions fn : Ω → G converges in (μ-)measure to
the function f : Ω → G (denoted fn → f (μ)) if for every ε > 0 we have
lim μ∗ ({ω ∈ Ω | fn (ω) − f (ω) ≥ ε}) = 0,
n→∞
with the notation μ∗ (E) = inf{μ(B) | B ∈ A, E ⊂ B} for all E ⊂ Ω.
Definition 4.3 A function f : Ω → G is (μ-)integrable if there is a sequence of
integrable simple functions fn : Ω → G such that
(a) fn → f (μ) and
(b) limm, n→∞ X |fm − fn |dμ = 0.
Above, we have denoted the function ω → g(ω) by |g| whenever g is a G-valued
function defined on Ω; we use this notation throughout this section. The condition
(b) can be expressed by saying that (fn ) is an L 1 Cauchy sequence. It can be shown
that in the situation of this definition the limit limn→∞ Ω fn exists and is independent
of the choice of the sequence of functions satisfying these conditions (see [1]). This
limit is called the (Bochner) integral of the μ-integrable function f (with respect
to μ) and denoted by Ω fdμ = Ω f (ω)dμ(ω). If f is a G-valued function defined
on a set containing E ⊂ Ω, then f is integrable over E if the function fE satisfying
/ E is integrable. We then write
fE (ω) = f (ω) for ω ∈ E and fE (ω) = 0 when ω ∈
f dμ =
E
f (ω) dμ(ω) =
E
Ω
fE dμ.
If E = Ω, it is often omitted in the notation. In these and similar notations we may
sometimes write μ(dω) instead of dμ(ω).