Appendix
A.1
The spectral theorem.
A.1.1 Herglotz’ theorem.
D EFINITION : A numerical sequence {an }n∈Z is positive definite if, for any choice
of (finitely many) complex numbers {zn }, we have
∑ an−mznzm ≥ 0.
(1.1)
n,m
Theorem (Herglotz). A numerical sequence {an }n∈Z is positive definite if, and
only if, there exists a positive measure µ ∈ M(T) such that an = µ̂(n) for all n.
P ROOF : Assume an = µ̂(n) with positive µ. Then
∑ an−mznzm = ∑
n,m
(1.2)
n,m
=
! "
!
e−int eimt zn zm dµ
"2
"
"
"∑ zn e−int " dµ ≥ 0.
n
Conversely, assume that {an } is positive definite. For arbitrary N and t ∈ T, choose
#
eint |n| ≤ N,
zn =
0
|n| > N.
We have ∑n,m an−m zn zm = ∑ j C j,N a j ei jt where C j,N is the number of ways to write
j in the form n−m where |n| ≤ N and |m| ≤ N, that is, C j,N = max(0, 2N +1−| j|).
It follows that
1
σ2N (S,t) =
C j,N a j ei jt ≥ 0,
2N + 1 ∑
j
$
%σ2N %L1 (T) = σN (t)dt = a0 . As N → ∞ the measures σN (t)dt converge in the
weak-star topology to a positive measure µ such that µ̂(n) = an for all n ∈ Z. !
61
Appendix
62
!1.1.2
If {an } is positive definite, then
(1.3)
|an | ≤ a0 ,
n+1
and the sequence {(an − an−1 +a
)} is positive definite.
2
This can be seen directly by checking condition (1.1), or deduced from Herglotz’s theorem and the following observations:
a. If µ is the positive measure such that an = µ̂(n), then a0 = %µ%, which
proves (1.3).
b. The measure ν = (1 − cost)µ is nonnegative, and its Fourier coeffin+1
cients are ν̂(n) = an − an−1 +a
. Also, since an = a−n , we have a0 − ℜa1 =
2
a−1 +a1
a0 − 2 = ν̂(0).
Combining all this, we obtain
Lemma. If {an } is positive definite, then
"%
an−1 + an+1 &""
"
(1.4)
" an −
" ≤ a0 − ℜa1 .
2
Positive definite functions can be defined over any abelian group by the same
inequality (1.1). A refinement of the lemma implies in particular that positive
definite functions on R that are continuous at 0 are in fact uniformly continuous.
1.1.3 Let H be a separable Hilbert space, and U a unitary operator on H . For
f ∈ H denote by H f the closed subspace of H spanned by {U n f }n∈Z .
Write an = 'U −n f , f (. The sequence {an } is positive definite since for any
finite sequence {zn } we have
∑ an−mznzm = ∑ 'znU −n f , zmU −m f (
n,m
n,m
(1.5)
'
'2
'
'
= '∑ znU −n f ' ≥ 0.
n
H
( f (n) =
By Herglotz’ theorem there exists a positive measure µ f ∈ M(T) such that µ
an . The measure µ f is called the spectral measure of f . Observe that
(1.6)
∑ an−mznzm = ∑
n,m
n,m
!
e−int eimt zn zm dµ =
)f (0) = % f %2 .
and in particular %µ f %M(T) = µ
JANUARY 12, 2012
! "
"2
"
−int "
z
e
"∑ n
" dµ ≥ 0.
n
A.1. The spectral theorem.
63
1.1.4 Comparing (1.6) and (1.5), one realizes that for finite sequences {zn } the
map ∑ znU n f )→ ∑ zn eint is an isometry into L2 (µ f ) and hence can be extended by
continuity to an isometric isomorphism
S : H f )→ L2 (µ f ).
(1.7)
S conjugates U H f with multiplication by eit on L2 (µ f ), and finite combinations
∑ znU n (restricted to H f ) with multiplication on L2 (µ f ) by ∑ zn eint ;
'
'
'
'
'∑ zn eint ' ∞ .
'∑ znU n '
=
(1.8)
L (µ )
L (H )
f
f
The conjugation extends, maintaining the isometry (1.8), to L (H f ,U), the norm
closure of the polynomials ∑ znU n in1 L (H ), to multiplications on L2 (µ f ) by the
continuous functions on the closed support of the measure µ f . It extends further
to conjugating the multiplications on L2 (µ f ) by functions belonging to L∞ (µ f ) to
elements of L (H f ,U), the closure in L (H f ) in the strong operator topology of
the set of finite sums {∑ znU n }.
1.1.5 Given µ, ν ∈ M(T), we use µ * ν to denote “µ is absolutely continuous
with respect to ν”.
Claim: If g ∈ H f then µg * µ f , and |Sg|2 is the Radon-Nikodym derivative.
P ROOF : If g ∈ H f then ϕ = Sg ∈ L2 (µ f ) and
(1.9)
so that µg = |ϕ|2 µ f .
(g (n) =
µ
!
e−int |ϕ|2 dµ f ,
Remark: Since SH f = L2 (µ f ), every positive ψ ∈ L1 (µ f ) is obtained as |Sg|2
for some g ∈ H f .
Claim: If g ∈ H f and µg ∼ µ f , (each absolutely continuous with respect to the
other,) then Hg = H f . Conversely, if Hg = H f then µg ∼ µ f .
P ROOF : The isometry S reduces the issue to the case in which H = L2 (µ f ), U
is the multiplication by eit , f = 1, and g ∈ L2 (µ f ). We have µg = |g|2 µ f , and the
assumption that µg ∼ µ f means g ,= 0 a.e. We need to show that the functions of
1 L (H
) is the space of bounded linear operators on H , endowed with the operator norm.
JANUARY 12, 2012
Appendix
64
the form Fg where F is a continuous function (or a trigonometric polynomial) are
$
dense in L2 (µ f ). If ψ is orthogonal to {Fg : F ∈ C(T)} then Fgψ̄dµ f = 0 for
all F so that gψ = 0 a.e. and since g ,= 0, we have ψ = 0.
Claim: If H f ⊥ Hg , then µ f +g = µ f + µg .
P ROOF : Since ' f ,U n g( = 'g,U n f ( = 0, we have
n
n
n
( f (n) + µ
(g (n).
!
µ
f +g (n) = '( f + g),U ( f + g)( = ' f ,U f ( + 'g,U g( = µ
Remarks: The condition H f ⊥ Hg follows from the seemingly weaker g ⊥ H f .
In fact if 'U n f , g( = 0 for all n, then 'U n f ,U m g( = 'U (n−m) f , g( = 0.
Given f , g ∈ H , by (1.8), the map ∑ an eint )→ '∑ anU n f , g( is a bounded linear
functional on C(T) so that there exists a (signed) measure µ f ,g such that
'U n f , g( = µ*
f ,g (n).
µ f ,g is absolutely continuous with respect to both µ f and µg ; it vanishes if, and
only if, H f ⊥ Hg .
Claim: There exist (finite or countable) sequences { fn } ⊂ H such that
(1.10)
H =
+
H fn
is an orthogonal decomposition. Moreover, any g ∈ H can be taken as f1 .
P ROOF : Let {un } be a dense subset of H . Take f1 = g. Take f2 in the orthogonal complement of H f1 such that u1 ∈ H f1 ⊕ H f2 . Take f3 in the orthogonal
complement of H f1 ⊕ H f2 such that u2 ∈ H f1 ⊕ H f2 ⊕ H f3 , etc.
1.1.6 A vector f ∈ H is said to be of maximal type for the system (H ,U) if
µg * µ f for every g ∈ H .
Clearly, if both f and g are of maximal type then µ f ∼ µg .
Claim: There exist f ∈ H of maximal type.
P ROOF : Let { fn } be as in 1.1.5 above, and an > 0 such that ∑ a2n % fn %2 < ∞. Write
f = ∑ an fn , then µ f = ∑ a2n µ fn . Given any g ∈ H , write g = ∑ gn with gn ∈ H fn
and observe that µg = ∑ µgn and µgn * µ fn , so that µg * µ f .
Remark: Given g ∈ H , if we take f1 = g in 1.1.5 we obtain f of maximal type
such that g ∈ H f .
JANUARY 12, 2012
A.1. The spectral theorem.
65
1.1.7 The spectral type of U is, by definition, the equivalence class M = {µ f }
of all µ f , such that f of maximal type. For notational convenience we choose and
fix some µ ∈ M .
Observe that the spectrum σ (U) of U is the µ-essential range2 of eit (that is
the range of eit on the closed support of µ).
1.1.8 Two Borel sets, E, G on T are equivalent mod µ (written E ∼ G mod µ)
if µ(E0G) = 0, equivalently, if 11E = 11G as elements of L∞ (µ).
For a Borel set E we write
(1.11)
HE = { f ∈ H : µ f carried by E}.
HE is clearly a closed subspace of H . The orthogonal projection onto HE correspond to the multiplication by 11E in the L2 model.
Claim: There exist a (possibly finite) sequence { fn } ⊂ H such that H fn are
,
mutually orthogonal, µ fn+1 * µ fn , and H = H fn .
The sequence can be chosen so that µ fn = 11En µ with Borel sets En satisfying
En+1 ⊂ En .
P ROOF : Take f1 ∈ H of maximal type. There is no loss of generality in assuming
µ f1 = µ. Write E1 = T. If H = H f1 we are done. Otherwise, let H 1 be the orthogonal complement of H f1 in H . H 1 is U-invariant and we take (temporarily)
for f2 any vector g of maximal type3 in H 1 .
Since µg * µ, we have µg = h(t)µ. Let E2 be a Borel set equivalent mod µ
to {t : h(t) > 0}, and observe that 11E2 ∈ L2 (µg ) and 11E2 µ ∼ µg . Choose f2 ∈ Hg
such that µ f2 = 11E2 µ. Observe that H f2 = Hg .
If H f2 = H 1 we are done, otherwise repeat the step in the orthogonal complement of H f2 in H 1 , etc. As in 6., use a dense sequence {un } to guarantee that
the union of H f j spans H .
1.1.9 D EFINITION : A Borel set E ⊂ T is of multiplicity ≥ n for (H ,U) if there
exist g1 , . . . , gn ∈ H such that µg j = 11E µ and Hg j ⊥ Hgk for j ,= k.
E is of multiplicity n if it is of multiplicity ≥ n, and no subset E 1 ⊂ E is of
multiplicity ≥ n + 1.
.
.
∞ X, B, µ , the spectrum of the opergenerally, if X, B, µ -is a measure
space,
F
∈
L
.
ator of multiplication by F on L2 X, B, µ is the essential range of F.
3 for the system (H 1 ,U
H 1)
2 More
JANUARY 12, 2012
Appendix
66
E is of (countably) infinite multiplicity if it is of multiplicity ≥ n for all n.
If E is a set of multiplicity 1 them (HE ,U) is linearly isometric to (L2 (µE ), Ũ)
(where Ũ is the operator of mulitplication by eit ), where µE = 11E µ.
If E is a set of multiplicity n then (HE ,U) is linearly isometric to an orthogonal direct sum of n copies of (L2 (µE ), Ũ), which is isometric to4 (L2 (µE ), Cn , Ũ).
If E is of infinite multiplicity, the successive selection described in 1.1.8, done
,
for (HE ,U), will give a sequence { fn } such that µ fn = µ, and HE = H f j is an
orthogonal direct sum of countably many copies of (L2 (µE ), Ũ). Equivalently, it
is isometric to (L2 (µE ), "2 ), Ũ), the space of "2 -valued measurable functions with
square summable norms.
Claim: In the notation of 1.1.8, the set G1 = E1 \ E2 is a set of multiplicity 1.
P ROOF : The vector f1 11G1 shows that the multiplicity is ≥ 1. If the multiplicity is
not equal to 1, there is a vector g2 ⊥ H f1 such that µg2 (G1 ) ,= 0; then the vector
g∗ = f2 + g2 11G1 is in H f⊥
, we have µ f2 * µg∗ but the two are not equivalent. This
1
contradicts the assumption that f2 was of maximal type in H f⊥
.
1
A repeated use of the same argument shows:
Claim: For every n, the set Gn = En \ En+1 is a set5 of multiplicity n.
Write E∞ = G∞ = ∩En , and µ∞ = 11G∞ µ.
The sets Gn are the biggest sets of multiplicity n and their union (including G∞ )
,
has full µ measure, thus H = HGn , and the decomposition does not depend
on the selection of the vectors fn . The function mU (t) = ∑∞
1 11En , equal to +∞ on
E∞ and to ∑∞
n
1
1
everywhere
else,
is
called
the
multiplicity
function
of (H ,U).
Gn
1
The spectral type and the multiplicity function are well defined and, together,
they are a complete invariant for (H ,U).
Theorem. Let H be a separable Hilbert space and U a unitary operator on H .
There exists in M(T) a sequence of mutually singular nonnegative measures µn ,
n ∈ N ∪ {∞} such that U is isometrically conjugate to the multiplication by eit on
,
the space6 L2 (µn , Cn ). The sequence {equivalence class of µn } is a complete
invariant.
4 That
is the space of Cn -valued measurable functions with square summable norms.
empty,
6 We abuse the notation by declaring that here C∞ means "2 .
5 possibly
JANUARY 12, 2012
A.2. Hilbert-Schmidt operators.
67
Example. A Bernoulli system {X, B, µ, T } consists of: X = {0, 1, . . . , r}Z that
is the space of all sequences x = {x j }, x j ∈ {0, 1, . . . , r}, j ∈ Z,
B is the σ -algebra spanned by the sets B j,a = {x : x j = a}, j ∈ Z, and a ∈
{0, 1, . . . , r}, µ is a product of copies of a probability measure p on {0, 1, . . . , r},
that is µ is defined by: for distinct ji ,
µ(x j1 = a1 , x j2 = a2 , . . . , x jm = am ) = p(a1 )p(a2 ) · · · p(am ).
(1.12)
Finally the measure-preserving transformation T in this system is the shift:
T ({xn }) = {xn+1 }.
The spectral measure µ f of a function f of integral zero which depends...
A.2
Hilbert-Schmidt operators.
Hilbert-Schmidt operators are essentially integral operators with square summable
kernel.
.
Theorem. Let H = L2 X, B, µ , µ ≥ 0. Hilbert-Schmidt operators on H are
precisely the integral operators with kernel K(x, y) in7 L2 (X × X), that is, the
operators k given by
(2.1)
k f (x) =
!
X
K(x, y) f (y)dµ(y).
P ROOF : Let K(x, y) ∈ L2 (X × X). By Fubini’s Theorem for a.e. x ∈ X
%K(x, ·)%2L2
=
!
|K(x, y)|2 dµ(y) < ∞
.
so that if f ∈ L2 X, B, µ , the integral (2.1) is well defined for a.e. x, and (by
Cauchy-Schwarz) is bounded by %K(x, ·)%L2 % f %L2 . Integrating the square of the
bound with respect to dµ(x) we obtain
(2.2)
%k f %L2 ≤ %K%L2 % f %L2 ,
so that k ∈ L H and %k%L H ≤ %K%L2 (X×X) .
.
To show that k is Hilbert-Schmidt, let { fn } is o.n. in L2 X, B, µ , and observe
that
(2.3)
7 L2 (X
∑|k fn(x)|2 ≤ %K(x, ·)%2L2 ;
× X) is a shorter notation for L2 (X × X), B × B, µ × µ)
JANUARY 12, 2012
Appendix
68
integrating (2.3) dµ(x) we obtain
h(k) ≤ %K%2L2 (X×X) .
(2.4)
.
2 X, B, µ . Let { f } be
Conversely, let -S be a Hilbert-Schmidt
operator
on
L
n
.
an o.n. basis for L2 X, B, µ . Define
K(x, y) = ∑ f¯n (y)S fn (x).
(2.5)
Then
(2.6)
!
2
|K(x, y)| dµ(y)dµ(x) =
!
∑|S fn(x)|2dµ(x) = h(S) < ∞,
.
so that K, as defined by (2.5), is in L2 (X × X). If g ∈ L2 X, B, µ , then
kg =
(2.7)
!
X
K(x, y)g(y)dµ(y) = ∑'g, fn (S fn = Sg.
!
The kernel K is independent of the choice of basis used in its definition, (2.5),
(since different kernels define different operators).
Lemma. Assume K ∈ L2 (X × X) and8 k = 0. then K = 0.
P ROOF : For any f , g ∈ L2 (X)
'K, f ⊗ g(L2 (X×X) = 'kḡ, f ( = 0.
(2.8)
but the functions f ⊗ g (x, y) = f (x)g(y) span L2 (X × X).
A.3
!
The Wold decomposition
Given a Hilbert space H1 , we denote by "2+ [H1 ] the unilateral "2 space with values
in H1 , that is, the space of sequences v = {xn }n≥0 such that xn ∈ H1 and %v%2 =
∑%xn %2 < ∞. If H1 = C, we write simply "2+ . As usual, "2 = "2 (Z) denotes the
space of bilateral square summable numerical sequences.
D EFINITION :
The standard shift S on "2+ [H1 ] is defined by
S({xn }n≥0 ) = {yn }n≥0 ,
8 The
operator k is defined by (2.1).
JANUARY 12, 2012
A.3. The Wold decomposition
69
where y0 = 0 and yn = xn−1 for n > 0.
A (unilateral, block) shift on H is an operator T ∈ L H which is unitarily
conjugate to the standard shift on some "2+ [H1 ].
Let Q be an isometry of H into H . If QH = H then (by definition) Q is
unitary.
If QH ,= H , write V1 = QH , W1 = (QH )⊥ , Vn = Qn H , and
Wn = Qn−1 W1 = Vn−1 ∩ Vn⊥ . Observe that Vn+1 ⊂ Vn , in fact Vn = Vn+1 ⊕ Wn .
,
Let V∞ = ∩n>0 Vn and W∞ = ∞
n=1 Wn then H = V∞ ⊕ W∞ , with both summands Q-invariant. Q is unitary on V∞ and its restriction to W∞ is a unilateral
block shift on "2+ [W1 ] (which is clearly isometric to W∞ ).
W1 is called the “wandering subspace of Q”.
When W1 is one dimensional, W∞ admits no reducing subspaces. In general
,
n−1 W ∗ where W ∗ is a
a reducing subspace of W∞ has the form W∞∗ = ∞
n=1 Q
closed subspace of W1
Theorem (Kolmogorov, von Neumann, Wold). Let Q be an isometry of H into
H . Then there is a unique decomposition H = V∞ ⊕ W∞ , with both summands
Q-invariant. Q is unitary on V∞ and its restriction to W∞ is a unilateral block
shift.
JANUARY 12, 2012
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