LOWER BOUNDS FOR EIGENVALUES OF SCHATTEN-VON NEUMANN OPERATORS

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Schatten-Von Neumann Operators M. I. Gil’

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LOWER BOUNDS FOR EIGENVALUES OF SCHATTEN-VON NEUMANN OPERATORS

M. I. GIL’

Department of Mathematics Ben Gurion University of the Negev P.0. Box 653, Beer-Sheva 84105, Israel EMail:gilmi@cs.bgu.ac.il

Received: 07 May, 2007

Accepted: 22 August, 2007

Communicated by: F. Zhang

2000 AMS Sub. Class.: 47A10, 47B10, 47B06.

Key words: Schatten-von Neumann ideals, Inequalities for eigenvalues.

Abstract: LetSpbe the Schatten-von Neumann ideal of compact operators equipped with the normNp(·). For anA∈Sp (1< p <∞), the inequality

"_∞ X

k=1

|Reλk(A)|^p

#¹_p +bp

"_∞ X

k=1

|Imλk(A)|^p

#¹_p

≥Np(AR)−bpNp(AI) (bp=const.>0)

is derived, whereλj(A) (j= 1,2, . . .)are the eigenvalues ofA,AI = (A− A^∗)/2iandAR = (A+A^∗)/2. The suggested approach is based on some relations between the real and imaginary Hermitian components of quasinilpotent operators.

Acknowledgements: 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.

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vol. 8, iss. 3, art. 66, 2007

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1. Statement of the Main Result

LetS_p (1≤ p <∞)be the Schatten-von Neumann ideal of compact operators in a separable Hilbert spaceHequipped with the norm

N_p(A) := [Trace(A^∗A)^p/2]^1/p<∞ (A∈S_p),

cf. [4, 6]. Letλj(A) (j = 1,2, . . .) be the eigenvalues ofA ∈ Sp taken with their multiplicities. In addition,σ(A)denotes the spectrum ofA,AI = (A−A^∗)/2iand A_R= (A+A^∗)/2are the Hermitian components ofA.

Recall the classical inequalities

j

X

k=1

|λ_k(A)|^p ≤

j

X

k=1

s^p_k(A) (p≥1, j = 1,2, . . .)

cf. [6, Corollary II.3.1] and

j

X

k=1

|Imλ_k(A)| ≤

j

X

k=1

s_k(A_I) (j = 1,2, . . .)

(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 (1.1) c_n=cn−1+

q

c²_n−1+ 1 (n= 2,3, . . .), c₁ = 1.

To formulate our main result, for ap∈[2ⁿ,2ⁿ⁺¹] (n = 1,2, . . .), put (1.2) bp =c^t_nc^1−t_n+1 with t= 2−2⁻ⁿp.

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For instance,b₂ =c₁ = 1,b₃ =√

c₁c₂ =p 1 +√

2≤1.554, b₄ =c₂ ≤2.415, b₅ =c^3/4₂ c^1/4₃ ≤2.900; b₆ = (c₂c₃)^1/2 ≤3.485; b₇ =c^1/4₂ c^1/4₃ ≤4.185 andb8 =c3 ≤5.027. In the case1< p <2,we use the relation

(1.3) b_p =bp/(p−1)

proved below.

The aim of this paper is to prove the following Theorem 1.1. LetA ∈S_p (1< p <∞). Then

(1.4)

" _∞ X

k=1

|Reλ_k(A)|^p

#¹_p +b_p

" _∞ X

k=1

|Imλ_k(A)|^p

#¹_p

≥N_p(A_R)−b_pN_p(A_I).

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)AbyiAwe get

Corollary 1.2. LetA ∈S_p (1< p <∞). Then

" _∞ X

k=1

|Imλ_k(A)|^p

#¹_p +b_p

" _∞ X

k=1

|Reλ_k(A)|^p

#¹_p

≥N_p(A_I)−b_pN_p(A_R).

Note that ifAis self-adjoint, then inequality (1.4) is attained, since

" _∞ X

k=1

|Reλ_k(A)|^p

#¹_p

=N_p(A_R) = N_p(A).

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Moreover, if A ∈ S₂ is a quasinilpotent operator, then from Theorem 1.1, it follows that N₂(A_R) ≤ N₂(A_I). However, as it is well known, N₂(A_R) = N₂(A_I), cf. [5, Lemma 6.5.1]. So in the case of a quasinilpotent Hilbert-Schmidt operator, inequality (1.4) is also attained.

Let{e_k}be an orthonormal basis inH, andF ∈S_pwithp≥2. Then by Theorem 4.7 from [3, p. 82],

N_p(F)≥

∞

X

k=1

kF e_kk^p

!¹_p

=





∞

X

k=1

" _∞ X

j=1

|f_jk|²

#^p₂



1 p

.

Herek · kis the norm inH andf_jk are the entries ofF in{e_k}. Moreover,

N_p(F)≤





∞

X

j=1

∞

X

k=1

|f_jk|^p⁰

!_p^p0



1 p

, 1 p+ 1

p⁰ = 1,

cf. [10, p. 298]. Leta_jk be the entries ofAin{e_k}. Then the previous inequalities yield the relations

N_p(A_R)≥m_p(A_R) :=





∞

X

k=1

∞

X

j=1

a_jk +a_kj 2

2!^p₂



1 p

and

Np(AI)≤Mp(AI) :=





∞

X

k=1

∞

X

j=1

a_jk−a_kj 2

p⁰!_p^p0



1 p

.

Now Theorem1.1implies:

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Corollary 1.3. LetA ∈S_p (2≤p <∞). Then

" _∞ X

k=1

|Reλ_k(A)|^p

#¹_p +b_p

" _∞ X

k=1

|Imλ_k(A)|^p

#¹_p

≥m_p(A_R)−b_pM_p(A_I).

Furthermore, from (1.1) it follows thatc_n+1 ≥2c_n ≥2ⁿ. Therefore, c_n+1 ≤c_n

1 +p

1 + 2^−(n−1)2 . Hence,

(1.5) c_n≤

n−1

Y

k=1

1 +p

1 + 4^−(k−1)

(n= 2,3, . . .).

Since √

1 +x≤1 + x

2, x∈(0,1), 1 +x≤e^x (x≥0), and

∞

X

k=1

1 4^k = 1

3, from inequality (1.5) it follows that

c_n+1 ≤2ⁿ

n

Y

k=1

(1 + 4^−k)≤2ⁿ⁺¹e^1/3 2 . Hence it follows that

(1.6) b_p ≤ pe^1/3

2 (2≤p < ∞).

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Indeed, by (1.2) forp=t2ⁿ+ (1−t)2ⁿ⁺¹ (n= 1,2, . . .; 0≤t ≤1)we have b_p =c^t_nc^1−t_n+1 ≤2^nt2^(1−t)(n+1)· e^1/3

2 = 2^n−t·e^1/3 2 . However,2^n−t≤p=t2ⁿ+ (1−t)2ⁿ⁺¹ (0≤t ≤1). So (1.6) is valid.

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2. Proof of Theorem 1.1

First let us prove the following lemma.

Lemma 2.1. Let V be a quasinilpotent operator, V_R = (V + V^∗)/2 and V_I = (V −V^∗)/2iits real and imaginary parts, respectively. Assume thatV_I ∈S₂ⁿ for an integern ≥2. ThenN2ⁿ(VR)≤cnN2ⁿ(VI).

Proof. To apply the mathematical induction method assume that for p = 2ⁿ there is a constant d_p, such that N_p(W_R) ≤ d_pN_p(W_I) for any quasinilpotent operator W ∈ S_p. Then replacing W by W i we have N_p(W_I) ≤ d_pN_p(W_R). Now let V ∈S_2p. ThenV² ∈S_p and therefore,

N_p((V²)_R)≤d_pN_p((V²)_I).

Here

(V²)_R= V²+ (V²)^∗

2 , (V²)_I = V²−(V²)^∗ 2i . However,

(V²)_R = (V_R)²−(V_I)², (V²)_I =V_IV_R+V_RV_I and thus

N_p(V_R²−V_I²)≤d_pN_p(V_RV_I +V_IV_R)≤2d_pN_2p(V_R)N_2p(V_I).

Take into account that

N_p((V_R)²) =N_2p² (V_R), N_p((V_I)²) = N_2p²(V_I).

So

N_2p² (VR)−N_2p² (VI)−2dpN2p(VR)N2p(VI)≤0.

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Solving this inequality with respect toN_2p(V_R), we get N_2p(V_R)≤N_2p(V_I)h

d_p+q

d²_p+ 1i

=N_2p(V_I)d_2p with

d_2p =d_p+q

d²_p+ 1.

In addition,d₂ = 1according to Lemma 6.5.1 from [5]. We thus have the required result withc_n=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 :S_p →S_r(1≤p, r <∞)is bounded if its norm

N_p→r(T) := sup

A∈Sp

N_r(T A) N_p(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 setπof projections inH is called a chain of projections if for allP₁, P₂ ∈ π eitherP₁ < P₂ orP₂ < P₁. This means that eitherP₁H ⊂ P₂H orP₂H ⊂ P₁H. A chain of projections is continuous if it does not have gaps. A continuous chain of projectionsπ is called a complete one if the zero and the unit operators belong toπ.

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 <· · ·< P_n=I, P_k ∈π, k= 1, . . . , n and an operatorR∈Sp put

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T_n=

n

X

k=1

P_kR∆P_k (∆P_k=P_k−Pk−1).

If there is a limitTn →T asn → ∞in the operator norm, we write T =

Z

π

P RdP.

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 anyR ∈S_p,1 < p < ∞. Due to Theorem I.6.1 [7], any Volterra operatorV withV_I ∈S_p can be represented as

V = 2i Z

π

P V_IdP.

Hence,

VR =Fπ(iVI), where

(2.1) F_π(R) :=

Z

π

P RdP + Z

π

P RdP ∗

(R ∈S_p,1< p <∞).

A transformer of this form is called a transformer of the triangular truncation with respect toπ.

Theorem III.6.3 from [7] asserts the following: Let πbe a complete continuous chain of projections inH. LetF_π(R)be a transformer of the triangular truncation with respect to π defined by (2.1). Then the norm N_p→p(F_π) is logarithmically convex. Moreover, the relation

(2.2) Np→p(F_π) =Nq→q(F_π)with 1 p+ 1

q = 1 (p≥2) is valid.

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Lemma 2.2. Let V be a quasinilpotent operator, and for a p ∈ [2ⁿ,2ⁿ⁺¹], n = 1,2, . . ., letV_I ∈S_p. Then

(2.3) N_p(V_R)≤b_pN_p(V_I).

Proof. By Lemma2.1, we have

N₂ⁿ→2ⁿ(F_π)≤c_n=b₂ⁿ. Put

p=t2ⁿ+ (1−t)2ⁿ⁺¹ (0≤t≤1).

Since the norm ofF_π is logarithmically convex andF_π(iV_I) =V_R, we can write Np→p(F_π)≤b^t₂nb^1−t₂n+1 (t = 2−2⁻ⁿp).

So N_p(V_R)

N_p(V_I) ≤b_p. This proves the lemma.

Furthermore, taking in (2.1)R = iV_I, by the previous lemma and the equalities (2.2) andF_π(iV_I) = V_R, we get

N_q(V_R)≤b_qN_q(V_I) (q ∈(1,2)) withbq=bp, q=p/(p−1). So we arrive at

Corollary 2.3. LetV ∈S_p be a quasinilpotent operator withp∈(1,2). Then (2.3) holds with (1.3) taken into account.

Proof of Theorem1.1. As it is well known, cf. [6] for any compact operatorA, there are a normal operatorDand a quasinilpotent operatorV, such that

(2.4) A =D+V andσ(D) = σ(A).

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Relation (2.4) is called the triangular representation of A; V andD are called the nilpotent part and diagonal one ofA, respectively. Clearly, by the triangular inequality,

N_p(V_R) =N_p(A_R−D_R)≥N_p(A_R)−N_p(D_R)

andN_p(A_I −D_I)≤N_p(A_I) +N_p(D_I). This and the previous lemma imply that N_p(A_R)−N_p(D_R)≤b_pN_p(A_I−D_I)≤b_p(N_p(A_I) +N_p(D_I)).

Hence,N_p(A_R)−b_pN_p(A_I)≤b_pN_p(D_I) +N_p(D_R). By (2.4), N_p^p(DR) =

∞

X

k=1

|Reλk(A)|^pandN_p^p(DI) =

∞

X

k=1

|Imλk(A)|^p.

So relation (1.4) is proved, as claimed.

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3. Additional Bounds

By Lemma 6.5.2 [5], for anA∈S₂ we have (3.1) N₂²(A)−

∞

X

k=1

|λk(A)|² = 2N₂²(AI)−2

∞

X

k=1

(Imλk(A))².

Hence,

N₂²(A)−

∞

X

k=1

|λ_k(A)|² = 2N₂²(A_R)−2

∞

X

k=1

(Reλ_k(A))²

and therefore,

N₂²(A_I)−

∞

X

k=1

(Imλ_k(A))² =N₂²(A_R)−

∞

X

k=1

(Reλ_k(A))².

Or

∞

X

k=1

(Reλk(A))²−

∞

X

k=1

(Imλk(A))² =N₂²(AR)−N₂²(AI) (A∈S2).

This equality improves Theorem 1.1 in the case p = 2. Moreover, from (3.1) it directly follows that

2

∞

X

k=1

(Imλ_k(A))² = 2N₂²(A_I)−N₂²(A) +

∞

X

k=1

|λ_k(A)|²

≥2N₂²(A_I)−N₂²(A) + TraceA². Now replacingAbyA^p we arrive at

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Theorem 3.1. LetA ∈S_2p (1≤p < ∞). Then

2

∞

X

k=1

(Im(λ^p_k(A)))² ≥2N₂²((A^p)_I)−N_2p^2p(A) + TraceA^2p.

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