The Schur-Horn Problem for Normal Operators

0.31 MB·English

by  Matthew Kennedy

THE SCHUR-HORN PROBLEM FOR NORMAL
OPERATORS
5
1
MATTHEW KENNEDY AND PAUL SKOUFRANIS
0
2

n Abstract. We consider the Schur-Horn problem for normal op-
a erators in von Neumann algebras, which is the problem of char-
J
acterizing the possible diagonal values of a given normal operator

6
based on its spectral data. For normal matrices, this problem is
2
well-known to be extremely difficult, and in fact, it remains open

] for matrices of size greater than 3. We show that the infinite
A dimensional version of this problem is more tractable, and estab-
O lish approximate solutions for normal operators in von Neumann
. factors of type I∞, II and III. A key result is an approximation
h
theoremthatcanbeseenasanapproximatemultivariateanalogue
t
a of Kadison’s Carpenter Theorem.
m
[

1
1. Introduction
v
7
In this paper we investigate the Schur-Horn problem for normal oper-
5
4 ators in von Neumann algebras. The general version of the Schur-Horn
6
problem is to characterize the possible diagonal values of a given opera-
0
. tor based on its spectral data. Since the appropriate notion of diagonal
1
0 for an operator in a von Neumann algebra is the conditional expecta-
5 tion of the operator onto a maximal abelian self-adjoint subalgebra
1
(MASA), the Schur-Horn problem can be formulated in the following
:
v
way.
i
X
Problem 1.1 (Schur-Horn). Let T be an operator in a von Neumann
r
a algebra M, and let be a MASA in M with corresponding conditional
expectation E : MA . Determine the elements of the set
A
→ A
(1) (T) := E (U∗TU) U a unitary in M .
A A
D { | }
The classical theorem of Schur [24] and Horn [10] completely solves
the Schur-Horn problem for self-adjoint matrices. A great deal of ef-
fort has gone into finding generalizations of this result for self-adjoint
2010 Mathematics Subject Classification. Primary 46L10; Secondary 15A42.
Key words and phrases. Schur-Horn,vonNeumannalgebra,MASA,eigenvalues,
diagonal, conditional expectation, normal operator.
First author partially supported by researchgrant from NSERC (Canada).
Second author partially supported by researchgrant from the NSF (USA).
1
2 M. KENNEDYAND P.SKOUFRANIS
operators in infinite-dimensional von Neumann algebras. We are par-
ticularly interested in the body of work inspired by the recent papers
of Neumann [20] and Arveson and Kadison [4].
For von Neumann factors of type II , the work contained in [1,2,
1
6,9] culminates in Ravichandran’s Schur-Horn theorem for self-adjoint
operators [23], which settles an open problem from [4]. For the state-
of-the-art results for von Neumann factors of type I , we direct the
∞
reader to the recent work of Kaftal and Weiss [14,15], Jasper [11], and
Bownik and Jasper [7,8].
The Schur-Horn problem for normal operators is seemingly substan-
tially more difficult than the corresponding problem for self-adjoint
operators. Indeed, for normal matrices the problem is known to be
equivalent to a difficult problem about the geometry of orthostochas-
tic matrices (cf. [3]), and little is known about the latter problem for
matrices of size greater than 3 (for matrices of size 3, see [5]).
Surprisingly, the Schur-Horn problem for normal operators is more
tractable in infinite dimensions. In this paper, we establish approxi-
mate Schur-Horn type theorems for normal operators in infinite dimen-
sional von Neumann algebras. Specifically, we establish a characteriza-
k·k
tion of the norm closure (N) of the set (N) in (1) when N is
A A
D D
a normal operator (satisfying certain assumptions, depending on the
context) in a von Neumann factor M of type I , II or III.
∞
It is not difficult to establish (cf. Lemma 4.1) that a necessary condi-
k·k
tion for an operator A to belong to (N) is that the spectrum σ(A)
A
D
of A belongs to the closed convex hull conv(σ(N)) of the spectrum of
N. If M is a factor of type I , then this condition turns out to be suf-
∞
ficient when is a continuous MASA. When is the discrete MASA,
A A
then we require an additional assumption on the essential spectrum of
N to circumvent the difficulties that arise for matrices.
Theorem 1.2. Let be a MASA in ( ) for an infinite-dimensional,
A B H
separable Hilbert space and let N be a normal operator in ( ). If
H B H
is continuous, then
A
k·k
(N) = A σ(A) conv(σ (N)) .
A e
D { ∈ A | ⊆ }
If is discrete, then the above equality also holds provided that σ(N)
A ⊆
conv(σ (N)), where σ (N) denotes the essential spectrum of N.
e e
When M is a II factor and N contains precisely three non-collinear
1
points, then an analogous result holds. Perhaps surprisingly, it turns
out (cf. Example 4.14) that this is the best result possible in this
setting.
THE SCHUR-HORN PROBLEM FOR NORMAL OPERATORS 3
Theorem 1.3. Let M be a type II factor with faithful normal tracial
1
state τ, and let be a MASA in M. Let N M be a normal operator
A ∈
such that σ(N) contains precisely three non-collinear points. Then
k·k
(N) = A τ(A) = τ(N),σ(A) conv(σ(N)) .
A
D { ∈ A | ⊆ }
We also establish similar results for normal operators in von Neu-
mann factors of type II and III, and for normal operators in Cuntz
∞
C*-algebras with two generators.
The key idea behind the above results is a theorem that can be
seen as an approximate multivariate analogue of Kadison’s Carpenter
Theorem (cf. [12,13]).
Theorem 1.4 (ApproximateMultivariateCarpenterTheorem). Let M
be a von Neumann factor of type I , II or III, and let be a MASA
∞
in M with corresponding conditional expectation E : AM . Let
A
A n be positive elements in such that n A = I →. SAuppose
{ k}k=1 A k=1 k M
that
P
M is separable if M is of type I , and
∞
• E is normal except possibly when M is type I and is con-
A ∞
• A
tinuous.
Then for every ǫ > 0 there are pairwise orthogonal projections P n
in M such that { k}k=1
n
(1) P = I ,
k=1 k M
(2) A E (P ) < ǫ,
k A k
(3) kPif M−is of typekI , then σ(P ) = σ (P ), and
∞ k e k
(4) if M is of type II, then τ(A ) = τ(P ).
K k
The proofof Theorem 1.4 isdivided into cases depending onthetype
of the von Neumann factor M. We also establish a similar result for
UHF C*-algebras.
The intractability of the Schur-Horn problem for normal matrices
can be explained (cf. Example 3.1) by the fact that there is no fi-
nite dimensional analogue of the approximate multivariate carpenter
theorem.
We note that Massey and Ravichandran [18] recently and indepen-
dently obtained results that are similar to the results in our paper. We
also mention the recent paper [17], where we establish a Schur-Horn
type theorem for arbitrary operators in a II factor by considering sin-
1
gular values instead of spectrum.
In addition to this introduction, there are three other sections. In
Section 2 we briefly review a required matricial result and develop
some useful terminology. In Section 3 we establish Theorem 1.4, the
4 M. KENNEDYAND P.SKOUFRANIS
approximate multivariate carpenter theorem. In Section 4 we establish
our results on the Schur-Horn problem for normal operators.
2. Background and Preliminaries
For n N, let (C) denote the n n matrices over C, let
n n
∈ M × D
denote the diagonal subalgebra, and let E : (C) denote the
n n n
M → D
conditional expectation of (C)onto . For T (C), tr(T) will
n n n
M D ∈ M
denote the (standard) trace of T, and τ(T) will denote the normalized
trace of T, i.e.
1
τ(T) = tr(T).
n
We require the following well-known and elementary result about
matrices. For the sake of completeness, we include a short proof.
Theorem 2.1. Let N (C) be a normal matrix. Then there exists
n
∈ M
a unitary operator U (C) such that every diagonal entry of U∗BU
n
∈ M
is τ(B) for all B C∗(N).
∈
Proof. Since N is a normal matrix, there exists a unitary W such that
W∗NW is diagonal. Fix B C∗(N). Note W∗BW is also diagonal.
2πi ∈
Let ζn = e n and let
(ζ1)1 (ζ2)1 ... (ζn−1)1 (ζn)1
n n n n
(ζ1)2 (ζ2)2 ... (ζn−1)2 (ζn)2
n n n n
1  . . . . 
V = .. .. .. .. .
√n
 (ζ1)n−1 (ζ2)n−1 ... (ζn−1)n−1 (ζn)n−1 
 n n n n 
 (ζ1)n (ζ2)n ... (ζn−1)n (ζn)n 
 n n n n 
 
It is easy to check that V is a unitary matrix and the diagonal entries
of the matrix V∗W∗BWV are equal to τ(B). Since B was arbitrary
we can take U = WV.
To simplify discussions of Theorem 1.4, we make the following defi-
nition.
Definition 2.2. Let N be a von Neumann algebra. A finite collection
of elements A n N is said to be a partition of unity in N if
{ k}k=1 ⊆ n
0 A I for all k 1,...,n and A = I .
≤ k ≤ N ∈ { } k=1 k N
3. Approximate MultivariatePCarpenter Theorem
In this section we will prove Theorem 1.4. The proof will be divided
intocases depending onthetype oftheunderlying vonNeumann factor.
But first, we will show that no analogue of Theorem 1.4 can hold in
finite dimensions.
THE SCHUR-HORN PROBLEM FOR NORMAL OPERATORS 5
3.1. Type I factors. The following example shows that no analogue
n
of Theorem 1.4 holds in finite dimensions, i.e. for von Neumann factors
of type I .
n
Example 3.1. In [3, Lemma 4], it is shown that if N = diag(0,1,i)
and A = diag 1, i, 1+i , then although σ(A) conv(σ(N)) there does
2 2 2 ⊆
not exist a unitary U (C) such that E (U∗NU) = A.
3 3
(cid:0) ∈(cid:1)M
Consider the projections
P = diag(1,0,0),P = diag(0,1,0), and P = diag(0,0,1),
1 2 3
along with the positive contractions
1 1 1 1 1 1
A = diag , ,0 ,A = diag ,0, , and A = diag 0, , .
1 2 3
2 2 2 2 2 2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Clearly these matrices all belong to , tr(A ) = 1 for k = 1,2,3 and
k
D
A +A +A = I .
1 2 3 3
HencebyKadison’sCarpenter TheoremthereareunitariesU ,U ,U
1 2 3
∈
(C) such that E (U∗P U ) = A for k = 1,2,3.
M3 3 k k k k
Now suppose that for any ǫ > 0 there is a unitary operator U
∈
(C) such that
3
M
A E (U∗P U) < ǫ
k 3 k
k − k
for k = 1,2,3. Then, since the unitary group of (C) is compact,
3
M
there must be a unitary U (C) such that A = E (U∗P U) for
3 k 3 k
∈ M
k = 1,2,3. However, writing N = 0P +1P +iP gives
1 2 3
1 i 1+i
E (U∗NU) = diag , , = A,
3
2 2 2
(cid:18) (cid:19)
which is a contradiction.
3.2. Type I factors. We now consider the case of the separable von
∞
Neumann factor of type I . This is the algebra ( ) of bounded
∞
B H
operators acting on a separable infinite dimensional Hilbert space .
H
3.2.1. The case of the discrete MASA. We first consider the case of the
(discrete) diagonal MASA in ( ). We will require the following
D B H
approximation result for partitions of unity in finite dimensions that
are nearly scalar operators.
Lemma 3.2. Let α n , β n [0,1] be such that n α =
1 = n β . The{n fko}rk=e1ve{rykǫ}k>=10⊆there exists ℓ N andkp=a1irwkise
orthogokn=a1l pkrojections P n (C) such that ∈n PP= I and
P { k}k=1 ∈ Mℓ k=1 k ℓ
E (P ) diag(α ,β ,β ,...,β ) < ǫ for all k.
ℓ k k k k k P
k − k
6 M. KENNEDYAND P.SKOUFRANIS
Proof. By applying a small perturbation, we may assume that α
k
and β are rational for all k. Fix ǫ > 0. Choose N ,N N and
k 1 2
∈
α′ n , β′ n N 0 such that α = 1 α′ and β = 1 β′ for
{ k}k=1 { k}k=1 ⊆ ∪ { } k N1 k k N2 k
all k. Choose N N such that
0
∈
α +N β′
k 0 k β < ǫ
k
N N +1 −
(cid:12) 0 2 (cid:12)
(cid:12) (cid:12)
for all k, and let ℓ = N (cid:12)+(N 1)N N (cid:12) N.
1 1 0 2
(cid:12) − (cid:12)∈
Notice that
(C) (C)⊕(N1−1)
MN1 ⊕MN0N2
can be identified with a matrix subalgebra of (C) such that
ℓ
M
⊕(N1−1) = .
DN1 ⊕DN0N2 Dℓ
Choose two collections of pairwise orthogonal projections
Q′ n and Q′′ n (C)
{ k}k=1 ⊆ DN1 { k}k=1 ⊆ DN0N2
such that
rank(Q′) = α′ and rank(Q′′) = N β′
k k k 0 k
for all k. Let
Q = Q′ (Q′′)⊕(N1−1).
k k ⊕ k
It is clear that Q n are pairwise orthogonal projections summing
{ k}k=1
to I .
ℓ
By Theorem 2.1 there exists a unitary W (C) such that each
1 ∈ MN1
diagonal entry of W∗Q′W is the average of the diagonal entries of Q′,
1 k 1 k
which is
1
(rank(Q′)) = α .
N k k
1
Thus the Q are simultaneously unitarily equivalent to projections of
k
the form
A (Q′′)⊕(N1−1)
k ⊕ k
where each diagonal entry of A (C) is α .
k ∈ MN1 k
For each j = 1,...,N 1, by pairing the (j+1)-th index in (C)
1− MN1
with the j-th matrix entry in the direct summand (C)⊕(N1−1),
MN0N2
we further obtain that the Q are jointly unitarily equivalent in (C)
k ℓ
M
to block matrices of the form
α A A A
k k,1,2 k,1,3 ··· k,1,N1
A D A A
 k,2,1 k k,2,3 ··· k,2,N1 
A A D A
 k...,3,1 .k.,3.,2 ..k. ·.·.·. k,...3,N1 
 
 A A A D 
 k,N1,1 k,N1,2 k,N1,3 ··· k 
 
THE SCHUR-HORN PROBLEM FOR NORMAL OPERATORS 7
where D (C) is a diagonal matrix with α appearing once
k ∈ MN0N2+1 k
along the diagonal, 1 appearing rank(Q′′) = N β′ times along the
k 0 k
diagonal, and all other diagonal entries being zero.
ApplyingTheorem2.1again,thereexistsaunitaryW (C)
2 ∈ MN0N2+1
suchthateachdiagonalentryofW∗D W istheaverageofthediagonal
2 k 2
entries of D , which is
k
α +N β′
k 0 k.
N N +1
0 2
It follows that conjugating the above matrices with the unitary
1 W⊕(N1−1) C (C),
⊕ 2 ∈ ⊕MN0N2+1
results in projection matrices P satisfying the desired conclusion.
k
Theorem 3.3. Theorem 1.4 holds in the case M = ( ) and is a
B H A
discrete MASA.
Proof. We begin with several perturbations to place the A s into a
k
more desirable form so that Lemma 3.2 may be applied. Since the
result is trivial for n = 1, we assume n 2. Fix ǫ > 0. Let e be
m m≥1
≥ { }
an orthonormal basis for corresponding to . By perturbing each
H D
A if necessary, we may assume (at a cost of 2ǫ to our approximation)
k
that 2ǫI A (1 2ǫ)I .
H n H
≤ ≤ −
Since a diagonal self-adjoint operator has at most countably many
eigenvalues of finite multiplicity, there is N N such that the diagonal
∈
entries of A with index greater than N are within ǫ of the essential
k n
spectrum of A for all k = 1,...,n 1. Since 2ǫI A (1 2ǫ)I ,
k H n H
− ≤ ≤ −
we may then, at a cost of ǫ to our approximation, perturb the diagonal
entries of each A so that for k = 1,...,n 1 the diagonal entries of
k
−
A with index greater than N belong to the essential spectrum of A
k k
and ǫI A (1 ǫ)I .
H n H
≤ ≤ −
By the above paragraphs, we have reduced the problem to the case
where ǫI A (1 ǫ)I and decomposes as = ′, where
H n H 0
≤ ≤ − H H H ⊕H
is a finite dimensional Hilbert space, such that with respect to this
0
H
decomposition of we can write
H
A = D A′,
k k ⊕ k
where D and A′ are diagonal operators satisfying
k k
n n
Dk = IH0, A′k = IH′, and σ(A′k) = σe(A′k) for k 6= n.
k=1 k=1
X X
Let e′ be the orthonormal basis of ′ obtained from e
{ m}m≥1 H { m}m≥1
by removing the first N elements and reindexing without reordering.
8 M. KENNEDYAND P.SKOUFRANIS
For k = 1,...,n 1 let α ℓk σ(A′) be an ǫ-cover. By per-
− { k,j}j=1 ⊆ k n
turbing both A′ and A′ simultaneously in each diagonal entry by at
k n
most ǫ, we may assume, at a cost of ǫ to the final approximation, and
n
of losing the inequality ǫIH′ ≤ A′n ≤ (1−ǫ)IH′, that
A′ = diag(β ,β ,...)
k k,1 k,2
where β α ℓk for k = 1,...,n 1.
k,m ∈ { k,j}j=1 −
For each (i ,...,i ) n−1 1,...,ℓ , let
1 n−1 ∈ k=1{ j}
I = j N β = α for all k = 1,...,n 1 .
(i1,...,in−1) { ∈ |Qk,j k,ij − }
Note that N is a disjoint union of the I . Therefore, by combin-
(i1,...,in−1)
ing elements of e′ according to which set I the index m
{ m}m≥1 (i1,...,in−1)
belongs to, we can write
= ′ ℓ
H H0 ⊕H0 ⊕ ⊕j=1Hj
where ℓ ∈ N, H0 and H0′ are finite dime(cid:0)nsional H(cid:1)ilbert spaces, each Hj
is an infinite dimensional Hilbert space, and such that with respect to
this decomposition of ,
H
A = D D′ ℓ γ I
k k ⊕ k ⊕ ⊕j=1 k,j Hj
where Dk and Dk′ are each diagonal o(cid:0)perators, (cid:1)
n n
Dk = IH0, Dj′ = IH′0,
k=1 k=1
X X
and
γ j = 1,...,ℓ,k = 1,...,n [0,1]
k,j
{ | } ⊆
n
with γ = 1 for each j. Note that although our assumptions
k=1 k,j
were only explicitlyA ,...A , we necessarily obtain the correct form
1 n−1
P n
for A because of the hypothesis that A = I .
n k=1 k H
Byincreasingℓifnecessary, bydecomposingoneoftheinfinitedimen-
P
sional Hilbert spaces occurring in the above decomposition into multi-
ple infinite dimensional Hilbert spaces, we may assume that dim(
0
H ⊕
′) ℓ. This decomposition into direct sums further reduces the
H0 ≤
problem to one of two cases: either
A = β I ( ) for all k,
k k K
∈ B K
where is a infinite dimensional Hilbert space and β [0,1] satisfy
k
n K ∈
β = 1, or
k=1 k
A = α β I (C ) for all k
P k k k K
⊕ ∈ B ⊕K
where isainfinitedimensional Hilbertspaceandα ,β [0,1]satisfy
k k
n K n ∈
β = α = 1.
k=1 k k=1 k
P P
THE SCHUR-HORN PROBLEM FOR NORMAL OPERATORS 9
For the first case, we may apply Lemma 3.2 with α = β to obtain
k k
m N and pairwise orthogonal projections P n (C) such
∈ n { k}k=1 ∈ Mm
that P = I , and such that each entry along the diagonal of
k=1 k m
P is within ǫ of β . In this case, the result now follows by taking an
k k
P
infinite direct sum of the P . To ensure that σ(P ) = σ (P ), we can
k k e k
perturbeach β so that 2ǫ β 1 2ǫand require the approximation
k k
≤ ≤ −
from Lemma 3.2 to be within ǫ of the desired matrices. This implies
that the projections with the desired diagonal are non-trivial. Thus,
taking their direct sum gives projections with spectrum equal to the
essential spectrum.
For the second case, we may apply Lemma 3.2 to obtain m N and
pairwiseorthogonalprojections P n (C)suchthat ∈n P =
{ k}k=1 ∈ Mm k=1 k
I , the first diagonal entry of each P is within ǫ of α , and such that
m k k
P
the other entries along the diagonal of P are each within ǫ of the cor-
k
responding β . The result in this case now follows from the first case
k
by excluding the first m basis vectors of .
K
3.2.2. The case of a continuous MASA. For the case of a continuous
MASA in ( ), we first deal with the case when each A is a scalar
k
B H
multiple of the identity.
Lemma 3.4. Theorem 1.4 holds in the case M = ( ), is a contin-
B H A
uous MASA, and A = α I for each k, where the α [0,1] satisfy
k k H k
n ∈
α = 1.
k=1 k
PProof. Without loss of generality, we may assume that αk = aNk for
each k, where the a N 0 satisfy n a = N. Since is a
k ∈ ∪ { } k=1 k A
continuous MASA, there exist pairwise orthogonal equivalent projec-
tions Q′ N such that N Q′ =PI . These projections give
{ j}j=1 ⊆ A j=1 j H
rise to a copy of (C) in ( ). Choose pairwise orthogonal pro-
N
M BPH
jections Q n such that each Q is a sum of a distinct Q′s.
{ k}k=1 ⊆ A k k j
Theorem 2.1 implies there exists a unitary U (C) ( ) such
N
∈ M ⊆ B H
that Q′U∗Q UQ′ = α Q′ for all j,k. Defining P = U∗Q U yields the
j k j k j k k
desired projections.
Theorem 3.5. Theorem 1.4 holds in the case M = ( ) and is a
B H A
continuous MASA.
Proof. By approximation, we may assume that the spectrum of each
A is finite. By taking compressions of all possible intersections of the
k
spectral projections of the A ’s (which are necessarily infinite projec-
k
tions since is continuous), we can apply Lemma 3.4 to each compres-
A
sion. Taking a sum of these finite number of compressions then yields
the result.
10 M. KENNEDYAND P.SKOUFRANIS
3.3. Type II factors. For the case of a von Neumann factor of type
1
II , we first consider thecasewhen eachA isapositive scalar. Forthis,
1 k
we require the following two technical results about approximating real
numbers.
Lemma 3.6. Given ǫ > 0, n N, and r n (0,1), there exists
∈ { k}k=1 ⊆
q n Q such that
{ k}k=1 ⊆
(1) q 1r ,r ,
k ∈ 2 k k
(2) rk qk < ǫ, and
Pnj=1(cid:2)rj − Pnj(cid:1)=1qj
(3) (cid:12)(cid:12)(cid:12)Pnjr=k1rj − Pnjr=k1−rqjk−(cid:12)(cid:12)(cid:12)qj < ǫ for all k
(cid:12) (cid:12)
Proof.(cid:12)Choose0 < δ min(cid:12) 1r k = 1,...,n andforeachk choose
(cid:12) ≤ (cid:12) 2 k
q 1r , 1r +δ Q. Then
k ∈ 2 k 2 k ∩ (cid:8) (cid:12) (cid:9)
(cid:12)
(cid:2) r (cid:1) 1r q 1r +δ r +2δ
k = 2 k k 2 k = k
2nδ + n r n 1r +δ ≤ n q ≤ n 1r n r
j=1 j j=1 2 k j=1 j j=1 2 j j=1 j
and P P (cid:0) (cid:1) P P P
r 2δ r 1r δ r q
k − = k − 2 k − k − k
n r n r 1r ≤ n r q
j=1 j j=1 j − 2 j j=1 j − j
r 1r
P P P k − 2 k
≤ n r 1r +δ
j=1 j − 2 j
r
k
= P (cid:0) (cid:0) (cid:1)(cid:1)
n
2nδ + r
− j=1 j
for each k. Therefore, since r are fixedP, the result follows by
k k≥1
{ }
choosing δ sufficiently small.
Lemma 3.7. Given ǫ > 0, n N, and (p )n (0,1) such that
∈ k k=1 ⊆
n p = 1, there exists q(k) j N,k = 1,...,n [0,1] Q such
k=1 k { j | ∈ } ⊆ ∩
that
P (k)
q
(k) j
q = p and p < ǫ
j k (cid:12) k − n q(k)(cid:12)
Xj≥1 (cid:12) k=1 j (cid:12)
(cid:12) (cid:12)
for all j N and for all k. (cid:12) P (cid:12)
∈ (cid:12) (cid:12)
Proof. Fix ǫ > 0. Applying Lemma 3.6 with r = p for all k, we
k k
obtain q(1),...,q(n) Q such that q(k) 1p ,p ,
1 1 ∈ 1 ∈ 2 k k
q(k) 1 (cid:2) p (cid:1) q(k) 1
p 1 < ǫ, and p k − 1 < ǫ.
(cid:12) k − n q(j)(cid:12) 2 (cid:12) k − n p q(j)(cid:12) 2
(cid:12) j=1 1 (cid:12) (cid:12) j=1 j − 1 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) P (cid:12) (cid:12) P (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)