Volume 8 (2007), Issue 3, Article 66, 7 pp.
LOWER BOUNDS FOR EIGENVALUES OF SCHATTEN-VON NEUMANN
OPERATORS
M. I. GIL’
D EPARTMENT OF M ATHEMATICS
B EN G URION U NIVERSITY OF THE N EGEV
P.0. B OX 653, B EER -S HEVA 84105, I SRAEL
gilmi@cs.bgu.ac.il
Received 07 May, 2007; accepted 22 August, 2007
Communicated by F. Zhang
A BSTRACT. Let Sp be the Schatten-von Neumann ideal of compact operators equipped with the
norm Np (·). For an A ∈ Sp (1 < p < ∞), the inequality
# p1
"∞
# p1
"∞
X
X
p
p
| Im λk (A)|
| Re λk (A)|
+ bp
≥ Np (AR ) − bp Np (AI ) (bp = const. > 0)
k=1
k=1
is derived, where λj (A) (j = 1, 2, . . . ) are the eigenvalues of A, AI = (A − A∗ )/2i and
AR = (A + A∗ )/2. The suggested approach is based on some relations between the real and
imaginary Hermitian components of quasinilpotent operators.
Key words and phrases: Schatten-von Neumann ideals, Inequalities for eigenvalues.
2000 Mathematics Subject Classification. 47A10, 47B10, 47B06.
1. S TATEMENT OF THE M AIN R ESULT
Let Sp (1 ≤ p < ∞) be the Schatten-von Neumann ideal of compact operators in a separable
Hilbert space H equipped with the norm
Np (A) := [Trace(A∗ A)p/2 ]1/p < ∞ (A ∈ Sp ),
cf. [4, 6]. Let λj (A) (j = 1, 2, . . . ) be the eigenvalues of A ∈ Sp taken with their multiplicities.
In addition, σ(A) denotes the spectrum of A, AI = (A − A∗ )/2i and AR = (A + A∗ )/2 are the
Hermitian components of A.
Recall the classical inequalities
j
X
k=1
p
|λk (A)| ≤
j
X
spk (A)
(p ≥ 1, j = 1, 2, . . . )
k=1
This research was supported by the Kamea fund of the Israel.
The author is very grateful to the referee for his very deep and helpful remarks.
148-07
2
M. I. G IL’
cf. [6, Corollary II.3.1] and
j
X
| Im λk (A)| ≤
k=1
j
X
sk (AI )
(j = 1, 2, . . . )
k=1
(see [6, Theorem II.6.1]). These results give us the upper bounds for sums of the eigenvalues
of compact operators. In the present paper we derive lower inequalities for the eigenvalues.
Our results supplement the very interesting recent investigations of the Schatten-von Neumann
operators, cf. [1, 2, 8, 9, 11, 12, 13, 14].
Let {cn }∞
n=1 be a sequence of positive numbers defined by
q
(1.1)
cn = cn−1 + c2n−1 + 1 (n = 2, 3, . . . ), c1 = 1.
To formulate our main result, for a p ∈ [2n , 2n+1 ] (n = 1, 2, . . . ), put
bp = ctn c1−t
with t = 2 − 2−n p.
n+1
p
√
√
For instance, b2 = c1 = 1, b3 = c1 c2 = 1 + 2 ≤ 1.554, b4 = c2 ≤ 2.415,
(1.2)
3/4 1/4
b5 = c 2 c 3
≤ 2.900;
b6 = (c2 c3 )1/2 ≤ 3.485;
1/4 1/4
b7 = c 2 c 3
≤ 4.185
and b8 = c3 ≤ 5.027. In the case 1 < p < 2, we use the relation
bp = bp/(p−1)
(1.3)
proved below.
The aim of this paper is to prove the following
Theorem 1.1. Let A ∈ Sp (1 < p < ∞). Then
"∞
# p1
"∞
# p1
X
X
(1.4)
| Re λk (A)|p + bp
| Im λk (A)|p ≥ Np (AR ) − bp Np (AI ).
k=1
k=1
The proof of this theorem is presented in the next section. Clearly, inequality (1.4) is effective
only if its right-hand part is positive.
Replacing in (1.4) A by iA we get
Corollary 1.2. Let A ∈ Sp (1 < p < ∞). Then
"∞
# p1
"∞
# p1
X
X
| Im λk (A)|p + bp
| Re λk (A)|p ≥ Np (AI ) − bp Np (AR ).
k=1
k=1
Note that if A is self-adjoint, then inequality (1.4) is attained, since
"∞
# p1
X
| Re λk (A)|p = Np (AR ) = Np (A).
k=1
Moreover, if A ∈ S2 is a quasinilpotent operator, then from Theorem 1.1, it follows that
N2 (AR ) ≤ N2 (AI ). However, as it is well known, N2 (AR ) = N2 (AI ), cf. [5, Lemma 6.5.1].
So in the case of a quasinilpotent Hilbert-Schmidt operator, inequality (1.4) is also attained.
Let {ek } be an orthonormal basis in H, and F ∈ Sp with p ≥ 2. Then by Theorem 4.7 from
[3, p. 82],
! p1  ∞ " ∞
# p2  p1
∞
X
X X
Np (F ) ≥
kF ek kp
=
|fjk |2  .
k=1
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 66, 7 pp.
k=1
j=1
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L OWER B OUNDS FOR E IGENVALUES OF S CHATTEN -VON N EUMANN O PERATORS
3
Here k · k is the norm in H and fjk are the entries of F in {ek }. Moreover,

! p0  p1
∞
∞
p
X
X
p0

 , 1 + 1 = 1,
Np (F ) ≤
|fjk |
p p0
j=1
k=1
cf. [10, p. 298]. Let ajk be the entries of A in {ek }. Then the previous inequalities yield the
relations
1

!p p
∞
∞ X
X
ajk + akj 2 2

Np (AR ) ≥ mp (AR ) := 
2
k=1
and
j=1

1
0! p p
∞
∞ X
X
ajk − akj p p0
 .
Np (AI ) ≤ Mp (AI ) := 
2
k=1
j=1
Now Theorem 1.1 implies:
Corollary 1.3. Let A ∈ Sp (2 ≤ p < ∞). Then
# p1
"∞
"∞
# p1
X
X
| Re λk (A)|p + bp
| Im λk (A)|p ≥ mp (AR ) − bp Mp (AI ).
k=1
k=1
Furthermore, from (1.1) it follows that cn+1 ≥ 2cn ≥ 2n . Therefore,
p
cn+1 ≤ cn 1 + 1 + 2−(n−1)2 .
Hence,
(1.5)
cn ≤
n−1
Y
p
1+
1+
4−(k−1)
(n = 2, 3, . . . ).
k=1
Since
√
x
1+x≤1+ ,
2
x ∈ (0, 1),
1 + x ≤ ex (x ≥ 0), and
∞
X
1
1
= ,
k
4
3
k=1
from inequality (1.5) it follows that
n
cn+1 ≤ 2
n
Y
(1 + 4−k ) ≤ 2n+1
k=1
e1/3
.
2
Hence it follows that
pe1/3
(2 ≤ p < ∞).
2
Indeed, by (1.2) for p = t2n + (1 − t)2n+1 (n = 1, 2, . . . ; 0 ≤ t ≤ 1) we have
(1.6)
bp ≤
e1/3
e1/3
= 2n−t ·
.
2
2
(0 ≤ t ≤ 1). So (1.6) is valid.
nt (1−t)(n+1)
bp = ctn c1−t
·
n+1 ≤ 2 2
However, 2n−t ≤ p = t2n + (1 − t)2n+1
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 66, 7 pp.
http://jipam.vu.edu.au/
4
M. I. G IL’
2. P ROOF OF T HEOREM 1.1
First let us prove the following lemma.
Lemma 2.1. Let V be a quasinilpotent operator, VR = (V + V ∗ )/2 and VI = (V − V ∗ )/2i
its real and imaginary parts, respectively. Assume that VI ∈ S2n for an integer n ≥ 2. Then
N2n (VR ) ≤ cn N2n (VI ).
Proof. To apply the mathematical induction method assume that for p = 2n there is a constant
dp , such that Np (WR ) ≤ dp Np (WI ) for any quasinilpotent operator W ∈ Sp . Then replacing
W by W i we have Np (WI ) ≤ dp Np (WR ). Now let V ∈ S2p . Then V 2 ∈ Sp and therefore,
Np ((V 2 )R ) ≤ dp Np ((V 2 )I ).
Here
(V 2 )R =
V 2 + (V 2 )∗
,
2
(V 2 )I =
V 2 − (V 2 )∗
.
2i
However,
(V 2 )R = (VR )2 − (VI )2 ,
(V 2 )I = VI VR + VR VI
and thus
Np (VR2 − VI2 ) ≤ dp Np (VR VI + VI VR ) ≤ 2dp N2p (VR )N2p (VI ).
Take into account that
2
2
(VI ).
Np ((VR )2 ) = N2p
(VR ), Np ((VI )2 ) = N2p
So
2
2
N2p
(VR ) − N2p
(VI ) − 2dp N2p (VR )N2p (VI ) ≤ 0.
Solving this inequality with respect to N2p (VR ), we get
q
h
i
N2p (VR ) ≤ N2p (VI ) dp + d2p + 1 = N2p (VI )d2p
with
d2p = dp +
q
d2p + 1.
In addition, d2 = 1 according to Lemma 6.5.1 from [5]. We thus have the required result with
cn = d2n .
We will say that a linear mapping T is a linear transformer if it is defined on a set of linear
operators and its values are linear operators. A linear transformer T : Sp → Sr (1 ≤ p, r < ∞)
is bounded if its norm
Nr (T A)
Np→r (T ) := sup
A∈Sp Np (A)
is finite. Below we give some examples of transformers. To prove relation (1.3) we need
Theorem III.6.3 from [7]. To formulate that theorem we recall some notions from [7, Section
I.3]. A set π of projections in H is called a chain of projections if for all P1 , P2 ∈ π either
P1 < P2 or P2 < P1 . This means that either P1 H ⊂ P2 H or P2 H ⊂ P1 H. A chain of
projections is continuous if it does not have gaps. A continuous chain of projections π is called
a complete one if the zero and the unit operators belong to π.
Let us introduce the integral with respect to a chain of projections π, cf. [7, Sections 1.4 and
I.5]. To this end for a partition
0 = P0 < P1 < · · · < Pn = I,
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 66, 7 pp.
Pk ∈ π, k = 1, . . . , n
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L OWER B OUNDS FOR E IGENVALUES OF S CHATTEN -VON N EUMANN O PERATORS
5
and an operator R ∈ Sp put
Tn =
n
X
(∆Pk = Pk − Pk−1 ).
Pk R∆Pk
k=1
If there is a limit Tn → T as n → ∞ in the operator norm, we write
Z
T = P RdP.
π
This limit is called the integral of R with respect to a chain of projections π. By Theorem III.4.1
from [7], this integral converges for any R ∈ Sp , 1 < p < ∞. Due to Theorem I.6.1 [7], any
Volterra operator V with VI ∈ Sp can be represented as
Z
V = 2i P VI dP.
π
Hence,
VR = Fπ (iVI ),
where
(2.1)
Fπ (R) :=
∗
Z
Z
P RdP +
π
(R ∈ Sp , 1 < p < ∞).
P RdP
π
A transformer of this form is called a transformer of the triangular truncation with respect to π.
Theorem III.6.3 from [7] asserts the following: Let π be a complete continuous chain of
projections in H. Let Fπ (R) be a transformer of the triangular truncation with respect to π
defined by (2.1). Then the norm Np→p (Fπ ) is logarithmically convex. Moreover, the relation
(2.2)
Np→p (Fπ ) = Nq→q (Fπ ) with
1 1
+ = 1 (p ≥ 2)
p q
is valid.
Lemma 2.2. Let V be a quasinilpotent operator, and for a p ∈ [2n , 2n+1 ], n = 1, 2, . . . , let
VI ∈ Sp . Then
Np (VR ) ≤ bp Np (VI ).
(2.3)
Proof. By Lemma 2.1, we have
N2n →2n (Fπ ) ≤ cn = b2n .
Put
p = t2n + (1 − t)2n+1 (0 ≤ t ≤ 1).
Since the norm of Fπ is logarithmically convex and Fπ (iVI ) = VR , we can write
−n
Np→p (Fπ ) ≤ bt2n b1−t
p).
2n+1 (t = 2 − 2
So
Np (VR )
≤ bp .
Np (VI )
This proves the lemma.
Furthermore, taking in (2.1) R = iVI , by the previous lemma and the equalities (2.2) and
Fπ (iVI ) = VR , we get
Nq (VR ) ≤ bq Nq (VI ) (q ∈ (1, 2))
with bq = bp , q = p/(p − 1). So we arrive at
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 66, 7 pp.
http://jipam.vu.edu.au/
6
M. I. G IL’
Corollary 2.3. Let V ∈ Sp be a quasinilpotent operator with p ∈ (1, 2). Then (2.3) holds with
(1.3) taken into account.
Proof of Theorem 1.1. As it is well known, cf. [6] for any compact operator A, there are a
normal operator D and a quasinilpotent operator V , such that
A=D+V
(2.4)
and σ(D) = σ(A).
Relation (2.4) is called the triangular representation of A; V and D are called the nilpotent part
and diagonal one of A, respectively. Clearly, by the triangular inequality,
Np (VR ) = Np (AR − DR ) ≥ Np (AR ) − Np (DR )
and Np (AI − DI ) ≤ Np (AI ) + Np (DI ). This and the previous lemma imply that
Np (AR ) − Np (DR ) ≤ bp Np (AI − DI ) ≤ bp (Np (AI ) + Np (DI )).
Hence, Np (AR ) − bp Np (AI ) ≤ bp Np (DI ) + Np (DR ). By (2.4),
Npp (DR )
=
∞
X
p
| Re λk (A)| and
Npp (DI )
=
∞
X
k=1
| Im λk (A)|p .
k=1
So relation (1.4) is proved, as claimed.
3. A DDITIONAL B OUNDS
By Lemma 6.5.2 [5], for an A ∈ S2 we have
∞
∞
X
X
2
2
2
(Im λk (A))2 .
|λk (A)| = 2N2 (AI ) − 2
(3.1)
N2 (A) −
k=1
k=1
Hence,
N22 (A)
−
∞
X
2
|λk (A)| =
2N22 (AR )
−2
k=1
∞
X
(Re λk (A))2
k=1
and therefore,
N22 (AI )
−
∞
X
2
(Im λk (A)) =
N22 (AR )
−
k=1
Or
∞
X
(Re λk (A))2 −
k=1
∞
X
∞
X
(Re λk (A))2 .
k=1
(Im λk (A))2 = N22 (AR ) − N22 (AI ) (A ∈ S2 ).
k=1
This equality improves Theorem 1.1 in the case p = 2. Moreover, from (3.1) it directly follows
that
∞
∞
X
X
2
(Im λk (A))2 = 2N22 (AI ) − N22 (A) +
|λk (A)|2
k=1
k=1
≥
2N22 (AI )
−
N22 (A)
+ Trace A2 .
Now replacing A by Ap we arrive at
Theorem 3.1. Let A ∈ S2p (1 ≤ p < ∞). Then
2
∞
X
2p
(Im(λpk (A)))2 ≥ 2N22 ((Ap )I ) − N2p
(A) + Trace A2p .
k=1
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 66, 7 pp.
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L OWER B OUNDS FOR E IGENVALUES OF S CHATTEN -VON N EUMANN O PERATORS
7
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J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 66, 7 pp.
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