Uncertainty principle and geometry of the infinite Grassmann manifold

0.13 MB·

by  Esteban Andruchow

Uncertainty principle and geometry of the infinite Grassmann
manifold
7
1
0
Esteban Andruchow and Gustavo Corach
2

n
January 16, 2017
a
J

3
1
Abstract

] We study the pairs of projections
A
F P f =χ f, Q f = χ fˆ ˇ, f L2(Rn),
h. I I J (cid:16) J (cid:17) ∈
at where I,J Rn are sets of finite Lebesgue measure, χI,χJ denote the corresponding char-
m acteristic fu⊂nctions andˆ,ˇdenote the Fourier-Planchereltransformation L2(Rn) L2(Rn)
→
[ and its inverse. These pairs of projections have been widely studied by several authors in
connection with the mathematical formulation of Heisenberg’s uncertainty principle. Our
1
study is done from a differential geometric point of view. We apply known results on the
v
Finsler geometry of the Grassmann manifold ( ) of a Hilbert space to establish that
3
P H H
3 there exists a unique minimal geodesic of ( ), which is a curve of the form
P H
7
3 δ(t)=eitXI,JPIe−itXI,J
0
1. which joins PI and QJ and has length π/2. As a consequence we obtain that if H is the
0 logarithm of the Fourier-Plancherelmap, then
7
1 [H,P ] π/2.
I
: k k≥
v
i The spectrum of XI,J is denumerable and symmetric with respect to the origin, it has a
X
smallest positive eigenvalue γ(X ) which satisfies
I,J
r
a
cos(γ(X ))= P Q .
I,J I J
k k
2010 MSC: 58B20, 47B15, 42A38, 47A63.
Keywords: Projections, pairs of projections, Grasmann manifold, uncertainty principle.
1 Introduction
Consider the following example:
Example 1.1. Let I,J Rn be Lebesgue-measurable sets of finite measure. Let P ,Q be the
I J
⊂
projections in L2(Rn,dx) given by
P f = χ f and Q f = χ fˆ ˇ,
I I J J
(cid:16) (cid:17)
1
where χ denotes the characteristic function of the set L. Equivalently, denoting by U the
L F
Fourier transformation regarded as a unitary operator acting in L2(Rn,dx) and by M the
ϕ
multiplication by ϕ, then
P = M and Q = U∗P U .
I χI J F J F
The operator P Q is Hilbert-Schmidt (see for instance [11], Lemma 2).
I J
An intuitive formulation of Heisenberg’s uncertainty principle says that a nonzero function
and its Fourier transform cannot be (simultaneously) sharply localized (see [13], page 207). We
give more precision to this statement below ( see for instance [11], page 906).
According to Folland and Sitaram [13], the idea of using projections P and Q to obtain a
I J
form of the uncertainty principle is due to Fuchs [14], and it was developed later in a series of
papers by Landau, Pollack and Slepian [20], [21], [25]. See the survey by Folland and Sitaram
[13].
DonohoandStark[11]provedthatifI,J Rn withfiniteLebesguemeasureandf L2(Rn)
⊂ ∈
with f = 1 satisfy that
2
k k
f(t)2dt < ǫ and fˆ(w)2dw < ǫ
I J
ZRn−I| | ZRn−J | |
then
I J (1 (ǫ +ǫ ))2.
I J
| || | ≥ −
Donoho and Stark showed several applications of these ideas to signal processing (and the
obstruction to the existence of an instantaneous frequency). Smith [26] generalized these results
to a locally compact abelian group G where I G and J Gˆ, the dual group of G. The
⊂ ⊂
books by Havin and Jo¨ricke [17], Hogan and Lakey [18], and Gro¨chenig [15] among many others,
contain further applications, generalizations and history of the different uncertainty principles.
By an elementary computation using Fubini’s theorem, Donoho and Stark prove that
P Q = I J ,
I J HS
k k | || |
p
where is Hilbert-Schmidt norm. Next they prove that
HS
k k
P Q 1 ǫ ǫ .
I J I J
k k ≥ − −
The fact that P Q P Q is well known.
I J I J HS
k k≤ k k
They argue that any bound c such that
P Q c < 1
I J
k k≤
is an expression of the uncertainty principle ([11], page 912).
Denote by ( ) the set of orthogonal projections of the Hilbert space , also called the
P H H
Grassmann manifold of . It is indeed a differentiable manifold of ( ) (also in the infinite
H B H
dimensional setting), with rich geometric structure (see for instance [24] or [7]). The pairs
(P ,Q ) might be put in the broader context of the sets
I J
= (P,Q) : P,Q are orthogonal projections and PQ is compact .
C { }
This set is a C∞-submanifold of ( ) ( ).
P H ×P H
2
An application of these geometrical results facts is a form of the uncertainty principle (see
Theorem 3.6 below).
Let us describe the content of the paper.
InSection 2werecalltheknownfacts onthegeometry of ( ). Insection 3weapplyknown
P H
results[24], [7],[2]ontheFinslergeometry oftheGrassmannmanifoldof tothespecialcaseof
H
pairs P ,Q . We prove that there exists a unique minimal geodesic of the Grassmann manifold
I J
of length π/2 which joins P and Q . That is, there exists a unique selfadjoint operator X of
I J I,J
norm π/2, which is co-diagonal with respect both to P and Q , such that
I J
eiXI,JP e−iXI,J = Q .
I J
The spectrum of the operator X is denumerable and symmetric with respect to the origin.
I,J
The smallest positive eigenvalue γ(X ) verifies
I,J
cos(γ(X )) = P Q .
I,J I J
k k
As a consequence from the fact that the minimal geodesic has length π/2, we prove that if
H is the logarithm of the Fourier transform in L2(Rn), and I Rn is a set of finite Lebesgue
⊂
measure, then
[H,P ] = [H,Q ] π/2.
I I
k k k k ≥
In Section 4 we show that for any pair of sets I,J Rn of finite measure, one has
⊂
N(P )+N(Q )= L2(Rn),
I J
where the sum is non-direct (the subspaces have infinite dimensional intersection).
2 Basic properties
2.1 Halmos decomposition
Let be a Hilbert space, ( ) the algebra of bounded linear operators in , ( ) the ideal
H B H H K H
of compact operators and ( ) the set of selfadjoint (orthogonal) projections, and ( ) the
∞
P H P H
subset of projections whose nullspaces and ranges have infinite dimension.
A tool that will be useful in the study of the pairs P ,Q is Halmos decomposition [16],
I J
which is the following orthogonal decomposition of : given a pair of projections P and Q,
H
consider
= R(P) R(Q) , = N(P) N(Q) , = R(P) N(Q) , = N(P) R(Q)
11 00 10 01
H ∩ H ∩ H ∩ H ∩
and the orthogonal complement of the sum of the above. This last subspace is usually called
0
H
the generic part of the pair P,Q. Note also that
N(P Q)= , N(P Q 1) = and N(P Q+1) = ,
11 00 10 01
− H ⊕H − − H − H
so that the generic part depends in fact of the difference P Q.
−
Halmos proved that there is an isometric isomorphism between and a product Hilbert
0
H
space such that inthe above decomposition (putting inplace of ), theprojections
0
L×L L×L H
are
1 0
P = 1 0 1 0
⊕ ⊕ ⊕ ⊕(cid:18) 0 0 (cid:19)
3
and
C2 CS
Q = 1 0 0 1 ,
⊕ ⊕ ⊕ ⊕(cid:18) CS S2 (cid:19)
where C = cos(X) and S = sin(X) for some operator 0 < X π/2 in with trivial nullspace.
≤ L
Aparently,thepair(P,Q)belongsto ifandonlyif isfinitedimensionalandC = cos(X)
11
C H
is compact.
Remark 2.1. If (P,Q) , then the spectral resolution of X can be easily described. Since
∈ C
0 < cos(X) is compact, it follows that
π
X = γ P + E,
n n
2
Xn
where 0 < γ < π/2 is an increasing (finite or infinite) sequence. For all n, dimR(P ) < , and
n n
∞
R(E) ( R(P )) = .
n≥1 n
⊕ ⊕ L
2.2 Finsler geometry of the Grassmann manifold of
H
Let us recall some basic facts on the differential geometry of the set ( ) (see for instance [7],
P H
[24], [2]).
1. The space ( ) is a homogeneous space under the action of the unitary group ( ) by
P H U H
inner conjugation: if U ( ) and P ( ), the action is given by
∈ U H ∈ P H
U P =UPU∗.
·
This action is locally transitive: it is well known that two projections P ,P such that
1 2
P P < 1, are conjugate. Therefore, since the unitary group ( ) is connected,
1 2
k − k U H
the orbits of the action coincide with the connected components of ( ), which are: for
P H
n N, ( ) (projections of nullity n), ( ) (projections of rank n) and ( )
n,∞ ∞,n ∞
∈ P H P H P H
(projectionsofinfiniterankandnullity). ThesecomponentsareC∞-submanifoldsof ( ).
B H
2. There is a natural linear connection in ( ). If dim < , it is the Levi-Civita connec-
P H H ∞
tion of the Riemannian metric which consists of considering the Frobenius inner product
at every tangent space. It is based on the diagonal / co-diagonal decomposition of ( ).
B H
To be more specific, given P ( ), the tangent space of ( ) at P consists of all self-
0 0
∈ P H P H
adjoint co-diagonal matrices (in terms of P ). The linear connection in ( ) is induced
0
P H
by a reductive structure, where the horizontal elements at P (in the Lie algebra of ( ):
0
U H
the space of antihermitian elements of ( )) are the co-diagonal antihermitian operators.
B H
The geodesics of which start at P are curves of the form
0
P
δ(t) = eitXP e−itX, (1)
0
with X∗ = X co-diagonal with respect to P . Observe that X is co-diagonal with respect
0
to every P = δ(t). It was proved in [24] that if P ,P ( ) satisfy P P < 1, then
t 0 1 0 1
∈ P H k − k
there exists a uniquegeodesic (up to reparametrization) joining P and P . This condition
0 1
is not necessary for the existence of a unique geodesic.
4
3. There exists a unique geodesic joining two projections P and Q if and only if
R(P) N(Q) = N(P) R(Q) = 0 ,
∩ ∩ { }
(see [2]).
4. If is infinite dimensional, the Frobenius metric is not available. However, if one endows
H
each tangent space of ( ) with the usual norm of ( ), one obtains a continuous (non
P H B H
regular) Finsler metric,
d(P ,P ) = inf ℓ(γ) :γ a continuous piecewise smooth curve in ( ) joining P and P
0 1 0 1
{ P H }
where ℓ(γ) denotes the length of γ (parametized in the interval I):
ℓ(γ) = γ˙(t) dt.
Z k k
I
In [24] it was shown that the geodesics (1) remain minimal among their endpoints for all
t such that
π
t .
| | ≤ 2 X
k k
It can be shown that d(P ,P ) < π/2 if and only if P P < 1. In other words,
0 1 0 1
k − k
P P = 1 if and only if d(P ,P ) = π/2.
0 1 0 1
k − k
3 Geometry of the pairs P , Q
I J
Lenard proved in [22] that the projections P ,Q (L2(Rn,dx)) defined in Example (1.1),
I J
∈ P
satisfy
R(P ) N(Q ) = R(Q ) N(P )= 0 . (2)
I J J I
∩ ∩ { }
Moreover, P Q = 1.
I J
k − k
Therefore one obtains the following:
Theorem 3.1. Let I,J be measurable subsets of Rn of finite measure, and P , Q the above
I J
projections. Then there exists a unique selfadjoint operator X satisfying:
I,J
1. X = π/2.
I,J
k k
2. X is P and Q co-diagonal. In other words, X maps functions in L2(Rn,dx) with
I,J I J I,J
support in I to functions with support in Rn I, and functions such that fˆhas support in
−
J to functions such that the Fourier transform has support in Rn J.
−
3. eiXI,JP e−iXI,J =Q .
I J
4. If P(t), t [0,1] is a smooth curve in ( ) with P(0) = P and P(1) = Q , then
I J
∈ P H
1
ℓ(P) = P˙(t) dt π/2.
Z k k ≥
0
5
Proof. By the condition (2) above ([22]), it follows from [2] that there exists a unique minimal
geodesic of ( ), of the form
P H
δ (t) =eitXI,JP eitXI,J
I,J I
with X∗ = X co-doagonal with respect to P (and Q ) such that
I,J I,J I J
δ (1) = Q .
I,J J
Condition 4. above is the minimality property of δ . Finally, the fact that P Q = 1
I,J I J
k − k
means that X = π/2.
I,J
k k
Remark 3.2. It is known [13] that λ = P Q P = P Q 2 < 1, and moreover √λ equals
1 I J I I J 1
k k k k
the cosine of the angle between the subspaces R(P ) and R(Q ).
I J
One can also relate this number λ with the operator X . Using Halmos decomposition
1 I,J
(recall that it consists only of and the generic part in this case),
00 0
H H
C2 0
P Q P = 0
I J I ⊕(cid:18) 0 0 (cid:19)
and thus λ = cos(X) 2. We shall see below that the spectrum of X is a strictly increasing
1
k k
sequence of positive eigenvalues γ π/2, with finite multiplicity. Moreover, since P Q P
n I J I
→
belongs to ( ), it follows that C ( ). Thus
1 2
B H ∈B L
cos(γ ) ℓ2.
n
{ } ∈
For a given P ( ), let be
P
∈ P H A
= X ( ):[X,P] is compact .
P
A { ∈ B H }
Apparently is a C∗-algebra.
P
A
Theorem 3.3. Let I,J be measurable subsets of Rn of finite Lebesgue measure.
1. The selfadjoint operator X has closed infinite dimensional range, in particular it is not
I,J
compact.
2. Let I be another measurable set with finite measure such that I I = 0, and let P = P .
0 | ∩ 0| 0 I0
Then, the commutant [X ,P ] is compact.
I,J 0
Proof. Easy matrix computations ([2]) show that, in the decomposition = ( ),
00
H H ⊕ L × L
X is of the form
I,J
0 iX
XI,J = 0⊕(cid:18) iX −0 (cid:19).
Note that the spectrum of this operator is symmetric with respect to the origin. Indeed, if V
equals the symmetry
0 1
V = 1 ,
⊕(cid:18) 1 0 (cid:19)
then apparently VX V = X . Also note that
I,J I,J
−
X2 0
X2 = 0 .
I,J ⊕(cid:18) 0 X2 (cid:19)
6
Therefore the spectrum of X is
I,J
σ(X ) = 0 γ : n 1 γ :n 1 ,
I,J n n
{ }∪{ ≥ }∪{− ≥ }
with 0 of infinite multiplicity, and the multiplicity of γ equal to the multiplicity of γ , and
n n
−
finite. What matters here, is that the set γ :n 1 is infinite, and is therefore an increasing
n
{ ≥ }
sequence converging to π/2. This holds because otherwise, the operator C would have finite
rank, and therefore P Q P would be of finite rank, which is not the case (see [22]). Thus X
I J I I,J
has closed range. of infinite dimension.
Note that P and Q satisfy that P P = 0 and Q P = Q P is compact, and therefore
I J I 0 J 0 J I0
P ,Q . ThusthesymmetriesS ,S belongtoa . SinceS = ei2XI,JS ,thisimplies
I J ∈ AP0 PI QJ P0 QJ PI
that
ei2XI,J .
∈ AP0
By the spectral picture of X it is clear that X can be obtained as an holomorphic function
I,J I,J
of ei2XI,J. Since is a C∗-algebra, this implies that X .
AP0 I,J ∈AP0
Let us relate the operator X with the mathematical version of the uncertainty principle,
I,J
according to [11] and [13].
Let A ( ) be an operator with closed range, the reduced minimum modulus γ of A is
A
∈ B H
the positive number
γ = min Aξ :ξ N(A)⊥, ξ = 1 = min λ :λ σ(A),λ = 0 .
A
{k k ∈ k k } {| | ∈ 6 }
Donoho and Stark [11] underline the role of the number Q P and consider any constant c
J I
k k
such that Q P c a manifestation of the (mathematical) uncertainty principle. By the
J I
k k ≤
above Remark, we have:
Corollary 3.4. With the current notations,
Q P =cos(γ ).
k J Ik XI.J
Proof. Indeed, in the above description of the spectrum of X , the reduced minimum modulus
I,J
γ of X coincides with γ .
XI.J I,J 1
Let X0 be the restriction of X to the generic part of P and Q , i.e., its restriction to
I,J I,J I J
N(X )⊥. In Halmos decomposition
I,J
0 iX
X0 = − .
I,J (cid:18) iX 0 (cid:19)
Recall the formula by Donoho and Stark [11]
P Q = I 1/2 J 1/2.
I J HS
k k | | | |
From the preceeding facts, it also follows:
Corollary 3.5. With the current notations
∞
1 1
I 1/2 J 1/2 = cos(X) = cos(X0 ) = cos(γ )2 1/2.
| | | | k kHS √2k I,J kHS { 2 n }
nX=1
7
Proof.
1 C2 0 1
I J = P Q 2 = Tr(P Q P )= Tr(C2) = Tr = Tr(cos(X0 )2).
| || | k I JkHS I J I 2 (cid:18) 0 C2 (cid:19) 2 I,J
This co-diagonal exponent X (with respect both to P and Q ) has interesting features
I,J I J
whenI = J and I < . Inthis case denoteby X = X ; then, wehave two unitaryoperators
I I,I
| | ∞
intertwining PI and QI. Namely, the Fourier transform UF and the exponential eiXI,
U∗P U = Q = eiXIP e−iXI.
F I F I I
Let H = H∗ be the natural logarithm of the Fourier transform, eiH = U . Namely, writing E ,
F 1
E , E and E the eigenprojections of U ,
−1 i −i F
π π
H = πE + E E .
−1 i −i
− 2 − 2
Note that H = π. Thus, one obtains a smooth path joining P and Q :
I I
k k
ϕ(t) = e−itHP eitH.
I
and, apparently, ϕ(1) = Q .
I
Since the Fourier transform intertwines P and Q , the norm of its commutant with either
I J
of these projections can be regarded as a measure of non commutativity between P and Q :
I J
Theorem 3.6. For any Lebesgue measurable set I Rn with I < , one has
⊂ | | ∞
[H,P ] = [H,Q ] π/2.
I I
k k k k ≥
Proof. The geodesic δ with exponent X is the shortest curve in ( ) joining P and Q . Its
I I I I
P H
length is π/2. Then
1 1
π/2 ℓ(ϕ) = ϕ˙(t) dt = eitH[H,P ]e−itH dt = [H,P ] .
I I
≤ Z k k Z k k k k
0 0
Note that
U∗[H,P ]U = [H,U∗P U ]= [H,Q ]
F I F F I F I
because U and H commute.
F
Remark 3.7.
1. We may write H in terms of U using the well known formulas
F
1 1 1
E = (1 U +U2 U3), E = (1 iU U2 +iU3), E = (1+iU U2 iU3),
−1 F F F i F F F −i F F F
4 − − 4 − − 4 − −
and thus
π
H = 1+(1+i)U U2 +(1+i)U3 .
F F F
4{− − }
Then
π
[H,P ] = (1+i)[U ,P ] [U2,P ]+(1+i)[U3,P ] .
I F I F I F I
4{ − }
The inequality in Corollary 3.6 can be written
(1+i)[U ,P ] [U2,P ]+(1+i)[U3,P ] 2.
F I F I F I
k − k ≥
8
2. In the special case when the set I is (essentially) symmetric with respect to the origin, P
I
commutes with U2, so that
F
[U2,P ]= 0 and [U3,P ] =[U ,P ]U2 = U2[U ,P ]
F I F I F I F F F I
one has
(1+i)π
[H,P ] = [U ,P ](1+U2).
I F I F
4
The operator U2f(x)= f( x) is a symmetry, then 1(1+U2) is the orthogonal projection
F − 2 F
E onto the the subspace of essentially even functions (f(x)= f( x) a.e.). Then one can
e
−
write
(1+i)π (1+i)π
[H,P ]= [U ,P ]E = E [U ,P ].
I F I e e F I
2 2
Corollary 3.8. Suppose that I is essentially symmetric, with finite measure.
1.
1
E [U ,P ] = E [U ,P ]E .
e F I e F I e
k k k k ≥ √2
2.
1
E P E Q ,
e I e I
k − k ≥ √2
where E P =P E and E Q = Q E are orthogonal projections.
e I I e e I I e
Proof. Recall that E and U commute. Then
e F
E [U ,P ]E =E (U P P U )E = U E (P U∗P U )E
e F I e e F I I F e F e I F I F e
− −
= U E (P Q )E .
F e I I e
−
where E , as well as U , and thus also Q = U∗P U commute with E .
e F I F I F e
The ranges of these two orthogonal projections E P and E Q consist of the elements of
e I e I
L2 which are essentially even and vanish (essentially) outside I, and the analogous subspace for
the Fourier transform.
4 Spatial properties of P and Q
I J
Let us return to the general setting (I not necessarily equal to J). The ranges and nullspaces
of P and Q have several interesting properties. First we need the following lemma:
I J
Lemma 4.1. Let P,Q be orthogonal projections such that P Q = 1. Then one and only
k − k
one of the following conditions hold:
1. N(P)+R(Q) = , with non direct sum (and this is equivalent to R(P)+N(Q) being a
H
direct sum and a closed proper subspace of ).
H
2. R(P)+N(Q) = , with non direct sum (and this is equivalent to N(P)+R(Q) being a
H
direct sum and a closed proper subspace of ).
H
9
3. R(P)+N(Q) is non closed (and this is equivalent to N(P)+R(Q) being non closed).
Proof. By the Krein-Krasnoselskii-Milman formula (see for instance [19])
P Q = max P(1 Q) , Q(1 P) ,
k − k {k − k k − k}
we have that one and only one of the following hold:
1. P(1 Q) < 1 and Q(1 P) = 1,
k − k k − k
2. P(1 Q) = 1 and Q(1 P) < 1, or
k − k k − k
3. P(1 Q) = 1 and Q(1 P) = 1.
k − k k − k
This alternative corresponds precisely with the three conditions in the Lemma. It is known [9]
that for two orthogonal projections E and F, EF < 1 holds if and only if R(E) R(F) = 0
k k ∩ { }
and R(E)+R(F) closed. The sum + of two subspaces is closed if and only if the sum
M N
⊥+ ⊥ is closed (see [9]). Therefore, EF <1 is also equivalent to N(E)+N(F) = .
M N k k H
If we apply these facts to E = P and F = 1 Q, we obtain that the first alternative is
−
equivalent to R(P) N(Q) = 0 and R(P)+N(Q) closed, or to N(P)+R(Q)= .
∩ { } H
Analogously, the second alternative is equivalent to R(Q) N(P) = 0 and R(Q)+N(P)
∩ { }
closed, or to N(Q)+R(P)= .
H
Note that in the first case, R(P)+ N(Q) is proper, otherwise its orthogonal complement
would be N(P) R(Q) = 0 , which together with the fact that N(P)+R(Q) = (closed!),
∩ { } H
would lead us to the second alternative.
Analogously in the second alternative, N(P)+R(Q) is proper.
If neither of these two happen, it is clear that neither R(P)+N(Q) nor (equivalently) the
sum of the orthogonals N(P)+R(Q) is closed.
We have the following:
Theorem 4.2. Let I,J Rn with finite Lebesgue measure. Then
⊂
1. R(P )+R(Q ) is a closed proper subset of L2(Rn), with infinite codimension. The sum is
I J
direct (R(P ) R(Q )= 0 ).
I J
∩ { }
2. N(P )+N(Q ) = L2(Rn), and the sum is not direct (N(P ) N(Q ) is infinite dimen-
I J I J
∩
sional).
3. R(P )+N(Q ) and N(P )+R(Q ) are proper dense subspaces of L2(Rn), and R(P )
I J I J I
∩
N(Q ) = N(P ) R(Q )= 0 .
J I J
∩ { }
Proof. By the cited result [9], two projections P,Q, satisfy that R(P) + R(Q) is closed and
R(P) R(Q)= 0 if and only if PQ < 1. It is also known (see above, [13]) that P Q < 1.
I J
∩ { } k k k k
Theintersection of these spaces is, in our case (usingthe notation of the Halmos decomposition)
R(P ) R(Q ) = = 0 .
I J 11
∩ H { }
Asremarkedabove,Lenardprovedthat = = = 0 ,and isinfinitedimensional.
11 10 01 00
H H H { } H
The orthogonal complement of this sum is
(R(P )+R(Q ))⊥ = N(P ) N(Q ) = .
I J I J 00
∩ H
10