Q-norm inequalities for sequences of Hilbert space operators

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This is the Published version of the following publication

Dragomir, Sever S, Moslehian, M and Sandor, J (2009) Q-norm inequalities for sequences of Hilbert space operators. Journal of Mathematical Inequalities, 3 (1). pp. 1-14. ISSN 1846-579X

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http://jmi.ele-math.com/03-01/Q-norm-inequalities-for-sequences-of-Hilbert-space- operators

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Inequalities Volume 3, Number 1 (2009), 1–14

Q–NORM INEQUALITIES FOR SEQUENCES OF HILBERT SPACE OPERATORS

S. S. DRAGOMIR, M. S. MOSLEHIAN ANDJ. S ´ANDOR

(communicated by R. Oinarov)

Abstract. We give some inequalities related to a large class of operator norms, the so-calledQ- norms, for a (not necessary commutative) family of bounded linear operators acting on a Hilbert space that are related to the classical Schwarz inequality. Applications for vector inequalities are also provided.

1. Introduction and preliminaries

Let (H;·,·) be a real or complex Hilbert space,B(H) the Banach algebra of all bounded linear operators acting onH andI denote the identity operator onH .

In many estimates one needs to use upper bounds for the norm of a sum of products A₁,...,A_n,B₁,...,B_n∈B(H), where separate information for norms of operators are provided. The special case in whichB_i=αiI(αi∈C,1in)and the norms in the question are the operator norm has already investigated in [4]. The celebrated H¨older’s discrete inequality stating

∑

n i=1αiβi

_n

i=1

∑

|αi|^p

_1/p _n

i=1

∑

|βi|^q _1/q

(1)

1<p, 1<q, 1 p+1

q=1, αi,βi∈C (1in)

and its variance in the special case p=q=2 (the so-called Cauchy–Bunyakovsky–

Schwarz (CBS) discrete inequality) are of special interest and of larger utility.

To each symmetric gauge functions defined on the sequences of real numbers there corresponds a unitarily invariant norm ||| · ||| defined on a two-sided ideal C_|||·||| of B(H) enjoying the invariant property |||UAV|||=|||A||| for all A∈C_|||·||| and all unitary operatorsU,V∈B(H). It is known that C_|||·||| is a Banach space under the norm||| · |||. If B∈C_|||·|||, then |||B^∗|||=|||B||| and |||AB|||A|||B||| for all A∈ B(H), from which we easily conclude that

|||ABC|||A|||B|||C (A,C∈B(H)).

Mathematics subject classification(2000): 47A05, 47A12.

Keywords and phrases: Bounded linear operators, Hilbert spaces, Schwarz inequality, Commuting operators.

c , Zagreb

Paper JMI-03-01 1

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Some well-known examples of unitarily invariant norms are the Schatten p-norms Bp:=tr(|B|^p)^1/p for 1p<∞, operator norm · , and the Ky-Fan norms; cf.

[1, 6]. A unitarily invariant norm · Q is called aQ-norm if there is a unitarily invariant norm · _Q such thatBB^∗_Q=B²_Q.

The aim of the present paper is to establish some upper bounds of interest for the quantity∑ⁿ_i=1A_iB_i_Q under certain assumptions onA_i’s andB_i’s. Our results are significant since we do not use any commutative assumption on the families of operators and as well we provide inequalities for a class of norms which are larger than operator norms (compare the results with those of [4]). Applications for vector inequalities are also given.

We need the following lemmas concerning real numbers (the second lemma is obvious):

LEMMA1.1. (Young’s inequality; see[5, p. 30 or p. 49]) If p>1and 1 p+1

q= 1,then

aba^p p +b^q

q for all a,b>0.

LEMMA1.2. If a,b>0, then ab1

4(a+b)²1

2(a²+b²).

LEMMA1.3. (Daykin–Eliezer; see[2]) If a_k>1,b_k>1 and 1 p+1

q <1,then _n

k=1

∑

a_kb_k ¹_p₊¹_q

_n

k=1

∑

a_k^p ¹_p _n

k=1

∑

b^q_k ¹_q

.

2. Some General Results

The following result containing 9 different inequalities may be stated:

THEOREM2.1. Let A₁,...,A_n∈B(H)and B₁,...,B_n∈CQ.Then

∑

n i=1

A_iB_i

2

Q

⎧⎨

⎩ A B C

(2)

(4)

where

A:=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

1knmaxAk² ∑ⁿ

i,j=1

B_iB^∗_j

Q,

1knmaxA_k _n

i=1∑A_i^r ¹_r

∑n i=1

n j=1∑

B_iB^∗_j

Q

_s¹_s , wherer>1, ¹_r+¹_s =1,

1knmaxA_k∑ⁿ

i=1A_imax

1in

n j=1∑

B_iB^∗_j_Q

,

B:=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

1inmaxA_i _n

k=1∑A_k^p ¹_p _n

i=1∑ n

j=1∑

B_iB^∗_j^q

Q

¹_q ,

_n

i=1∑Ai^t ¹_t _n

k=1∑ Ak^p ¹_p⎡

⎣∑ⁿ

i=1

n j=1∑

BiB^∗_j^q

Q

^u_q⎤

⎦

1u

, wheret>1, ¹_t +¹_u=1,

∑n i=1Ai

_n

k=1∑ Ak^p ¹_p

1inmax

⎧⎨

⎩ n

j=1∑

BiB^∗_j^q_Q ¹_q⎫

⎬

⎭,

(3)

for p>1, 1 p+1

q =1; and

C:=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

1inmaxAi ∑ⁿ

k=1Ak∑ⁿ

i=1max

1jn

B_iB^∗_j

Q

, _n

i=1∑Ai^m _m¹ _n

k=1∑ Ak n

i=1∑

1maxjn

BiB^∗_j_Q _l¹_l

, wherem,l>1, _m¹+¹_l =1, _n

k=1∑ Ak ₂

1i,maxjn

BiB^∗_j

Q

.

Proof. In the operator partial order ofB(H), we have 0

n i=1

∑

BiAi

_∗

∑

n i=1

BiAi

(4)

=

∑

ⁿ

i=1

A^∗_iB^∗_i

∑

ⁿ

j=1

BjAj=

∑

ⁿ

i,j=1

A^∗_iB^∗_iBjAj.

(5)

Taking the norm in (4) and noticing thatBB^∗_Q=B²_Q for anyB∈CQ, we have

∑

n i=1

A_iB_i

2

Q

=

∑

n i=1

∑

n j=1

A^∗_iB^∗_iB_jA_j

Q

∑

ⁿ

i=1

∑

n

j=1AiA_jB_iB^∗_j

Q

=

∑

ⁿ

i=1A_i n

∑

j=1

A_jB_iB^∗_j

Q

=:M.

Utilizing H¨older’s discrete inequality we have that

∑

n j=1

AjBiB^∗_j

Q

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1knmaxAk ∑ⁿ

j=1

BiB^∗_j

Q, _n

k=1∑A_k^p ¹_p

∑n j=1

B_iB^∗_j^q

Q

¹_q , wherep>1, ¹_p+¹_q=1,

∑n

k=1A_k max

1jn

B_iB^∗_j

Q,

for anyi∈ {1,...,n}.

This provides the following inequalities:

M

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1knmaxAk ∑ⁿ

i=1Ai n

j=1∑

B_iB^∗_j

Q

=:M₁,

_n

k=1∑ Ak^p ¹_p _n

i=1∑Ai n

j=1∑

BiB^∗_j^q

Q

¹_q :=Mp, wherep>1, ¹_p+¹_q=1,

∑n

k=1Ak ∑ⁿ

i=1Ai

1maxjn

BiB^∗_j

Q

:=M∞.

(6)

Utilizing H¨older’s inequality forr,s>1, 1 r+1

s =1,we have:

∑

n i=1

Ai n

∑

j=1

B_iB^∗_j

Q

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

1inmaxA_i ∑ⁿ

i,j=1

B_iB^∗_j

Q, _n

i=1∑A_i^r ¹_r

∑n i=1

n j=1∑

B_iB^∗_j_Q _s¹_s

, wherer>1, ¹_r+¹_s =1,

∑n

i=1Aimax

1in

n j=1∑

B_iB^∗_j

Q

, and thus we can state that

M₁

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

1knmaxAk² ∑ⁿ

i,j=1

BiB^∗_j_Q,

1knmaxAk _n

i=1∑Ai^r ¹_r

∑n i=1

n j=1∑

BiB^∗_j_Q s¹_s

, wherer>1, ¹_r+¹_s =1,

1knmaxAk ∑ⁿ

i=1Aimax

1in

n j=1∑

B_iB^∗_j

Q

, and thefirst part of the theorem is proved.

By H¨older’s inequality we can also have that (for p>1, 1 p+1

q =1)

M_p n

k=1

∑

A_k^p ¹_p

×

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

1inmaxAi∑ⁿ

i=1

n j=1∑

BiB^∗_j^q_Q ¹_q

,

_n

i=1∑Ai^t ¹_t ⎡

⎣∑ⁿ

i=1

n j=1∑

B_iB^∗_j^q

Q

^u_q⎤

⎦

1u

, wheret>1, ¹_t +¹_u=1,

∑n

i=1Aimax

1in

⎧⎨

⎩ n

j=1∑

BiB^∗_j^q

Q

¹_q⎫

⎬

⎭, and the second part of (2) is proved.

(7)

Finally, we may state that

M_∞

∑

ⁿ

k=1A_k ×

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1inmaxAi ∑ⁿ

i=1max

1jn

BiB^∗_j_Q

, _n

i=1∑A_i^m _m¹

∑n i=1

1jnmax

B_iB^∗_j

Q

_l¹_l , wherem,l>1, _m¹+¹_l =1,

∑n

i=1Ai max

1i,jn

B_iB^∗_j

Q

, giving the last part of (2).

REMARK2.2. One can obtain various particular inequalities as well. For instance, by using the usual Cauchy–Bunyakovsky–Schwarz inequality and norm prop- erties of the spaceCQ we get

∑

n i=1

AiBi

2

Q

n

i=1

∑

AiBi_Q ₂

n

i=1

∑

A_iB_i_Q ¹₂

∑

ⁿ

i=1

Ai²

∑

ⁿ

i=1

Bi²_Q. (5)

If we consider again the Cauchy–Bunyakovsky–Schwarz inequality

∑

n i=1

A_iB_i

2

Q

∑

ⁿ

i=1Ai²_Q

∑

ⁿ

i=1Bi²_Q (6)

then we can conclude that (5) is a refinement of the (CBS) inequality (6) whenever the operator norm · is majorized by · Q.

COROLLARY2.3. Let A₁,...,A_n∈B(H)and B₁,...,B_n∈CQ so that B_iB^∗_j=0 with i=j.Then

∑

n i=1

AiBi

2

Q

⎧⎨

⎩ A˜ B˜ C˜

(7)

(8)

where

A˜:=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

1knmax Ak² ∑ⁿ

i=1Bi²_Q,

1knmax Ak _n

i=1∑Ai^r ¹_r _n

i=1∑Bi^2s_Q ¹_s

, wherer>1, ¹_r+¹_s =1,

1knmax A_k∑ⁿ

i=1A_imax

1in

B_i²_Q ,

B˜:=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

1inmaxAi _n

k=1∑ Ak^p ¹_p _n

i=1∑Bi²_Q, _n

i=1∑Ai^t ¹_t _n

k=1∑ Ak^p ¹_p _n

i=1∑Bi^2u_Q ¹_u

, wheret>1, ¹_t +¹_u=1,

∑n i=1A_i

_n

k=1∑ A_k^p ¹_p

1inmax

B_i²_Q , where p>1and

C˜:=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1inmaxA_i ∑ⁿ

k=1A_k ∑ⁿ

i=1B_i²_Q, _n

i=1∑A_i^m _m¹ _n

k=1∑A_k _n

i=1∑B_i^2l_Q ¹_l

, wherem,l>1, _m¹+¹_l =1, _n

k=1∑A_k 2

1i,maxjn

B_i²_Q . As in the proof of the Theorem 2.1 one has

∑

n i=1

AiBi

2

Q

∑

ⁿ

i=1Ai _n

∑

j=1AjBiB^∗_j_Q

=:M

Using H¨older’s generalized inequality from Lemma 1.3, and assuming thatAj>

1, BiB^∗_j_Q>1,we get _n

∑

j=1AjBiB^∗_j_Q

_n

k=1

∑

Ak^p

_p^q₊_q _n

∑

j=1BiB^∗_j^q_Q _p^P₊_q

. Thus the following result is true:

(9)

THEOREM2.4. Let A₁,...,An∈B(H)and B₁,...,Bn∈CQ. Assume thatAj

>1andBiB^∗_j_Q>1 for all i,j∈ {1,2,...,n}. Then

∑

n i=1

A_iB_i

2

Q

n

k=1

∑

A_k^p _p^q

+q n i=1

∑

A_i

n

∑

j=1B_iB^∗_j^q_Q _p^p

+q

,

where 1 p+1

q<1.

3. Other Results

A different approach is embodied in the following theorem:

THEOREM3.1. If A₁,...,A_n∈B(H)and B₁,...,B_n∈CQ,then

∑

n i=1

A_iB_i

2

Q

∑

ⁿ

i=1A_i²

∑

ⁿ

j=1

B_iB^∗_j

Q (8)

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∑n

i=1Ai²max

1in

n j=1∑

BiB^∗_j_Q

,

_n

i=1∑Ai^2p ¹_p

∑n i=1

n j=1∑

B_iB^∗_j

Q

_q¹_q , wherep>1, ¹_p+¹_q=1,

1inmaxAi² ∑ⁿ

i,j=1

BiB^∗_j

Q.

Proof. From the proof of Theorem 2.1 we have that

∑

n i=1

A_iB_i

2

Q

∑

ⁿ

i=1

∑

n j=1

AiA_jB_iB^∗_j

Q.

Using the simple observation that AiA_j1

2

Ai²+A_j²

, i,j∈ {1,...,n},

(10)

we get

∑

n i=1

∑

n

j=1AiA_jB_iB^∗_j

Q1

2

∑

n i=1

∑

n j=1

Ai²+A_j²B_iB^∗_j

Q

=1 2

∑

n i=1

∑

n j=1

A_i²B_iB^∗_j

Q+A_j²B_jB^∗_i

Q

=

∑

ⁿ

i=1Ai²

∑

ⁿ

j=1

BiB^∗_j

Q, which proves thefirst inequality in (8).

The second part follows by H¨older’s inequality and the details are omitted.

REMARK3.2. If in (8) we chooseA₁=...=An=I,then we get

∑

n i=1

Bi

Q

_n

i,

∑

j=1

BiB^∗_j

Q

_1/2

_n

i,

∑

j=1Bi_QB^∗_j

Q

_1/2

=

∑

n i=1

Bi

Q

,

which is a refinement for the generalized triangle inequality for · _Q, whenever · Q

is majorized by · _Q.

The following corollary may be stated:

COROLLARY3.3. If A₁,...,A_n∈B(H)and B₁,...,B_n∈CQ are such that B_iB^∗_j

=0for i=j, i,j∈ {1,...,n},then

∑

n i=1

A_iB_i

2

Q

∑

ⁿ

i=1Ai²Bi²_Q (9)

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∑n

i=1Ai²max

1inBi²_Q, _n

i=1∑Ai^2p ¹_p

∑n j=1Bi^2q_Q

¹_q , wherep>1, ¹_p+¹_q=1,

1inmaxAi²∑ⁿ

i=1Bi²_Q.

(11)

THEOREM3.4. If A₁,...,An∈B(H)and B₁,...,Bn∈CQ,then

∑

n i=1

A_iB_i

2

Q

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

1inmaxAi² ∑ⁿ

i,j=1

B_iB^∗_j

Q, _n

i=1∑A_i^p ²_p

∑n i,j=1

B_iB^∗_j^q_Q ¹_q

, wherep>1, ¹_p+¹_q=1, _n

i=1∑Ai ₂

1i,maxjn

BiB^∗_j_Q

.

(10)

Proof. We know that

∑

n i=1

A_iB_i

2

Q

∑

ⁿ

i=1

∑

n

j=1A_iA_jB_iB^∗_j

Q=:P. Firstly, we obviously have that

P max

1i,jn AiA_j!

∑

ⁿ

i,j=1

B_iB^∗_j

Q= max

1inAi²

∑

ⁿ

i,j=1

B_iB^∗_j

Q. Secondly, by the H¨older inequality for double sums, we obtain

P n

i,j=1

∑

"

AiAj#^p¹_p

∑

n i,j=1

BiB^∗_j^q

Q

¹_q

= n

i=1

∑

Ai^p

∑

ⁿ

j=1

A_j^p¹_p

∑

n i,j=1

B_iB^∗_j^q

Q

¹_q

= n

i=1

∑

A_i^p ²_p

∑

n i,j=1

B_iB^∗_j^q

Q

¹_q , wherep>1, 1

p+1 q =1. Finally, we have

P max

1i,jn

B_iB^∗_j

Q

n

i,j=1

∑

AiA_j

= n

i=1

∑

A_i ₂

1i,maxjn

B_iB^∗_j

Q

and the theorem is proved.

(12)

COROLLARY3.5. If A₁,...,An∈B(H)and B₁,...,Bn∈CQ are such that BiB^∗_j

=0for i,j∈ {1,...,n}with i=j, then

∑

n i=1

A_iB_i

2

Q

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1inmaxA_i² ∑ⁿ

i=1B_i²_Q, _n

i=1∑A_i^p ²_p _n

i=1∑B_i^2q_Q ¹_q

, wherep>1, ¹_p+¹_q=1, _n

i=1∑Ai ₂

1inmax

Bi²_Q .

(11)

THEOREM3.6. If 1 p+1

q <1 andA_i>1 andB_iB^∗_j>1 for all 1i,jn

then

∑

n i=1

A_iB_i

2

Q

_n

i=1

∑

Ai^p

_p^2q₊_q _n

i,

∑

j=1BiB^∗_j^q_Q _p₊^p_q

.

Proof. Using Lemma 1.3 for double sums, we have, as in the proof of Theorem 3.4, that

p=

∑

ⁿ

i=1

∑

n

j=1A_iA_jB_iB^∗_jQ n

i,

∑

j=1(A_iA_j)^p _p+q^q

∑

n

i,j=1B_iB^∗_j^q_Q _p+q^p

, if we assume thatA_i>1 and B_iB^∗_j>1 for all 1i,jn.

Thus, we get

p n

i=1

∑

A_i^p _p+q^2q

∑

n

i,j=1B_iB^∗_j^q_Q _p+q^p

, if 1

p+1 q<1.

Finally, the following result may be stated as well:

THEOREM3.7. If 1 p+1

q =1, A₁,...,A_n∈B(H )and B₁,...,B_n∈CQ, then

∑

n i=1

AiBi

2

Q

1

p

∑

n

i=1Ai^p+1 q

∑

n i=1Ai^q

1inmax n

∑

j=1BiB^∗_jQ

.

(13)

Proof. As in the proof of Theorem 3.1 we have

∑

n i=

A_iB_i

2

Q

∑

ⁿ

i=1

∑

n j=1

AiAjBiB^∗_j_Q.

Now, using Lemma 1.1, we can write AiAj 1

pAi^p+1 qAj^q. Thus, we have

∑

n i=1

∑

n

j=1A_iA_jB_iB^∗_jQ

∑

ⁿ

i=1

∑

n j=1

1

pA_i^p+1 qA_j^q

B_iB^∗_jQ

= 1 p

∑

n i=1

∑

n

j=1Ai^pBiB^∗_j_Q+1 q

∑

n i=1

∑

n

j=1Aj^qBiB^∗_j_Q

= 1 p

∑

n i=1

Ai^p

∑

ⁿ

j=1

BiB^∗_j_Q+1 q

∑

n j=1

Aj^q

∑

ⁿ

i=1

BiB^∗_j_Q

1 p

∑

n

i=1A_i^p+1 q

∑

n j=1A_j^q

1inmax n

∑

j=1B_iB^∗_jQ

.

REMARK3.8. Another variant may be obtained via Lemma 1.2:

A_iA_j1

4(A_i+A_j)²,so

∑

n i=1

∑

n

j=1AiAjBiB^∗_j_Q1 4

∑

n i=1

∑

n

j=1(Ai+Aj)²BiB^∗_j_Q

sup

1in

∑

n j=1

BiB^∗_j_Q _n

i,

∑

j=1

(Ai+Aj)²

.

4. Applications

Throughout this section we assume · Q (and hence · _Q) to be the operator norm. If byM(B,A)we denote any of the bounds provided by (2), (5), (8) or (10) for the quantity

∑ⁿ

i=1A_iB_i

², then we may state the following general fact:

Under the assumptions of Theorem 2.1, we have:

∑

n i=1

A_iB_ix

2

x²M(B,A). (12)

(14)

for any x∈H and

∑

n

i=1AiBix,y

2

x²y²M(B,A). (13)

for any x,y∈H, respectively.

The proof follows by the Schwarz inequality in the Hilbert space (H,.,.). Now, we consider the non zero vectorsy₁,...,yn∈H . Define Bi:H →H by Bi=^y_yⁱ^⊗y_iⁱ (1in). Then

B_iB^∗_j=y_j,y_iy_i⊗y_j

yiyj =y_i,y_j (1i,jn).

If (y_i)_1in is an orthogonal family on H, then B_i=1 and B_iB_j=0 for i,j∈ {1,...,n}, i=j.

Now, utilizing, for instance, the inequalities in Theorem 3.1 we may state that:

∑

n i=1

A_ix,y_i y_i y_i

2

x²

∑

ⁿ

i=1Ai²

∑

ⁿ

j=1

$y_i,y_j% (14)

x²×

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∑n

i=1A_i²max

1in

n j=1∑

$y_i,y_j% ,

_n

i=1∑Ai^2p ¹_p

∑n i=1

n j=1∑

$yi,yj%q¹_q , wherep>1, ¹_p+¹_q=1,

1inmaxAi² ∑ⁿ

i,j=1

$yi,yj%. for anyx,y1,...,yn