Norm for Sums of Two Basic Elementary Operators

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Volume 2012, Article ID 643568,12pages doi:10.1155/2012/643568

Research Article

Norm for Sums of Two Basic Elementary Operators

A. Bachir,

1, 2, 3

_{F. Lombarkia,}

1, 2, 3

_{and A. Segres}

1, 2, 3

1_{Department of Mathematics, Faculty of Science, King Khalid University, P.O. Box 9004,}

Abha, Saudi Arabia

2_{Department of Mathematics, Faculty of Science, University of Batna, 05000 Batna, Algeria} 3_{Department of Mathematics, Mascara University, Algeria}

Correspondence should be addressed to A. Bachir,bachir [email protected]

Received 27 July 2012; Revised 8 November 2012; Accepted 25 November 2012

Academic Editor: Yuri Latushkin

Copyrightq2012 A. Bachir et al. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We give necessary and suﬃcient conditions under which the norm of basic elementary operators

attains its optimal value in terms of the numerical range.

1. Introduction

LetEbe a normed space overKRorC,SEits unit sphere, andE∗its dual topological space. LetDbe the normalized duality mapping formEtoE∗given by

Dx ϕ∈E∗_:_ϕ_x _x2_,_ϕ_x_, _∀_x_∈_E. _1.1

LetBEbe the normed space of all bounded linear operators acting onE. For any operator

A∈BEandx∈E,

WxA ϕAx:ϕ∈Dx,

WA ∪{WxA:x∈SE} 1.2

is called the spatial numerical range ofA, which may be defined as

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This definition was extended to arbitrary elements of a normed algebraAby Bonsall1–3

who defined the numerical range ofa∈ Aas

Va WAa, 1.4

whereAais the left regular representation ofAinBA, that is,Aaabfor allb∈ A.Va is known as the algebra numerical range ofa ∈ A, and, according to the above definitions,

Vais defined by

Va ϕab:b∈SA; ϕ∈Db. 1.5

For an operator A ∈ BE, Bachir and Segres 4 have extended the usual definitions of numerical range from one operator to two operators in diﬀerent ways as follows.

The spatial numerical rangeWA_BofA∈BErelative toBis

WA_BϕAx:x∈S_E; ϕ∈DBx. 1.6

The spatial numerical rangeGA_BofA∈BErelative toBis

GA_BϕAx:x∈E; Bx1, ϕ∈DBx. 1.7

The maximal spatial numerical range ofA∈BErelative toBis

MA_BϕAx:x∈SE;BxB, ϕ∈DBx. 1.8

ForA, B∈BE, letSEB {xn_n:xn∈SE,Bxn → B}, then the set

MA_BlimϕnAxn:xn_n∈SEB, ϕn∈DBxn 1.9

is called the generalized maximal numerical range ofArelative toB. It is known thatMA_B is a nonempty closed subset ofKandMA_B ⊆ MA_B ⊆WA_B. The definition ofMA_B can be rewritten, with respect to the semi-inner product·,·as

MA_B{limAxn, Bxn:xn_n∈SEB}, 1.10

with respect to an inner product·,·as

MA_B{limAx_n, Bx_n:x_n_n∈S_EB}. 1.11

We shall be concerned to estimate the norm of the elementary operator MA1,B1 MA2,B2,

where A1, A2, B1, B2 are bounded linear operators on a normed spaceE and MA1,B1 is the

basic elementary operator defined onBEby

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We also give necessary and suﬃcient conditions on the operatorsA1, A2, B1, B2under which

MA1,B1 MA2,B2attaints its optimal valueA1B1 A2B2.

2. Equality of Norms

Our next aim is to give necessary and suﬃcient conditions on the set {A1, A2, B1, B2} of

operators for which the norm ofMA1,B1 MA2,B2equalsA1B1 A2B2.

Lemma 2.1. For any of the operatorsA, B, C∈BEand allα, β∈K, one has

MαA βB_BαMA_B βB2_;

MαA βC_B⊆αMA_B βMC_B. 2.1

Proof. The proof is elementary.

Theorem 2.2. LetA1, A2, B1, B2be operators inBE.

IfA1A2 ∈ MA1A2∪ MA2A1andB1B2 ∈ MB1B2∪ MB2B1, then

MA1,B1 MA2,B2A1B1 A2B2. 2.2

Proof. The proof will be done in four steps; we choose one and the others will be proved

similarly. Suppose that A1A2 ∈ MA1_A₂ and B1B2 ∈ MB1_B₂, then there exist

xnn ∈ SEA2, ϕn ∈DA2xnsuch thatA1A2 limnϕnA1xnand there existynn ∈ SEB2, ψn∈DB2ynsuch thatB1B2limnψnB1yn. Define the operatorsXn∈BEas

follows:

Xnynψn⊗xnynψnynxn, ∀n. 2.3

ThenXn ≤ B2,for alln≥1, and

_M_A₁_B

1 MA2 B2Xn

ynA1XnB1 A2XnB2yn

A1XnB1yn A2XnB2yn

ϕn

_A₁_ψn

B1yn

xn A2ψn

B2yn

xn

≥ 1

ϕnϕn

ψnB1ynA1xn ψnB2ynA2xn

1

ϕnψn

B1ynϕnA1xn B2yn2A2xn2.

2.4

MA1,B1 MA2,B2 ≥

_M_A

1,B1 MA2,B2Xn

yn

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Hence

MA1,B1 MA2,B2 ≥

ψnB1ynϕnA1xn B2yn2A2xn2

A2B2 , ∀n≥1. 2.6

Lettingn → ∞,

MA1,B1 MA2,B2 ≥ A1B1 A2B2. 2.7

Since

MA1,B1 MA2,B2 ≤ A1B1 A2B2, 2.8

therefore

MA1,B1 MA2,B2A1B1 A2B2. 2.9

Corollary 2.3. LetEbe a normed space andA, B∈BE. Then, the following assertions hold:

1ifAB ∈ MA_B, thenA BA B;

2ifA ∈ MI_AandB ∈ MI_B, thenM_A,B I1 AB.

Remark 2.4. In the previous corollary, if we setB I, then we obtain an important equation

called the Daugavet equation:

A I1 A. 2.10

It is well known that every compact operator onC0,1 5or onL10,1 6satisfies2.10.

A Banach spaceEis said to have the Daugavet property if every rank-one operator on

Esatisfies2.10. So that from our Corollary2.3if 1∈ MI_Aor 1∈ MA_Ifor every rank-one operatorA, thenEhas the Daugavet property.

The reverse implication in the previous theorem is not true, in general, as shown in the following example which is a modification of that given by the authors Bachir and Segres4, Example 3.17.

Example 2.5. Letc0 be the classical space of sequencesxnn ⊂ C : xn → 0, equipped with

the normx_n_n max_n|x_n|and letLbe an infinite-dimensional Banach space. Taking the Banach spaceE L⊕c0 equipped with the norm, forx x1 x2 ∈ E,x x1 x2

max{x1,Tx1 x2}, whereTis any norm-one operator fromLtoc0which does not attain

its normby Josefson-Nissenzweig’s theorem7, we can find a sequenceϕ_n_n ⊂S_E∗such thatϕnconverges weakly to 0. Therefore we get the desired operatorT :L → c0defined by

Tx_n _nn

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LetA1, A2, B1, B2be operators defined onEas follows:

A1x1 x2 0 Tx1;

A2xA2x1 x2 x1 0;

B1x1 x2 x1−x2;

B2I, ∀x x1 x2∈L×c0,

2.12

where I is the identity operator on E. It easy to check thatA1, A2, B1 are linear bounded

operators andA1A2B1B21. If we chooseX0Iandx0 x1 0 such that

1Tx1 ≥ x1, thenX0x01 and

MA1,B1 MA2,B2 ≥ MA1,B1 MA2,B2X0x0

A1X0B1 A2X0B2x0

0 Tx1 x1 0

max{x1,2Tx1}

2,

2.13

and from

MA1,B1 MA2,B2 ≤ A1A2 B1B22 2.14

we get

MA1,B1 MA2,B22A1A2 B1B2. 2.15

It is clear from the definitions ofMA1A2andWA1A2that

MA1A2⊆WA1A2 2.16

for details, see4.

The next result shows that the reverse is true under certain conditions, before that we recall the definition of Birkhoﬀ-James orthogonality in normed spaces.

Definition 2.6. LetEbe a normed space andx, y ∈E. We say thatxis orthogonal toyin the

sense of Birkhoﬀ-James8,9, in shortx⊥_B₋_Jy, iﬀ

∀λ∈K:x λy≥ x. 2.17

IfF, Gare linear subspaces ofE, we say thatFis orthogonal toGin the sense of⊥B−J,

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If T ∈ BE, we will denote by RanT and T† the range and the dual adjoint, respectively, of the operatorT.

Theorem 2.7. LetA1, A2, B1, B2be operators inBE.

IfMA1,B1 MA2,B2A1B1 A2B2,

Ran A†₂⊥B−JRan

A†₁−A_A1₂A†₂

, RanB2⊥B−J Ran

B1−B_B1₂B2

, 2.18

then

A1A2 ∈ M A†₁

A†

2

, B1B2 ∈ MB1B2. 2.19

Moreover, if

Ran A†₁⊥B−JRan

A†₂−A2

A1A†1

, RanB1⊥B−J Ran

B2−B_B2₁B1

, 2.20

then

A1A2 ∈ M A†₁

A†

2

∩ M A†₂

A†

1

, B1B2 ∈ MB1_B₂∩ MB2_B₁. 2.21

Proof. IfMA1,B1 MA2,B2A1B1 A2B2, then we can find two normalized sequences

Xnn⊆BEandxnn⊆Esuch that

lim

n A1XnB1xn A2XnB2xnA1B1 A2B2. 2.22

We have for alln≥1

A1XnB1xn ≤ A1B1xn ≤ A1B1

A2XnB2xn ≤ A2B2xn ≤ A2B2,

2.23

so we can deduce from the above inequalities and 2.10 that limnB1xn B1 and

limnB2xnB2. From the assumptions RanB2⊥B−J RanB1−B1/B2B2we get

Ran

B1−B_B1₂B2

∩RanB2 {0}. 2.24

Setχn B1−B1/B2B2xnand yn B2xnfor allnand define the functionφnon the

closed subspaceFspanned by{x_n, y_n}for allnas

φnaχn bynbyn2_b_B

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It is clear thatφnis linear for allnand

_φn

aχn byn|b|B2xn2aχn bynB2xn

_by_n

_aχ_n _by_n. 2.26

From the assumptions RanB2⊥B−J RanB1−B1/B2B2it follows that

_φ

aχn byn≤ B2xnaχn byn, ∀a, b∈K,∀n. 2.27

This means thatφ_nis continuous for eachnon the subspaceFwithφ_nB2xnby2.27

andφnyn ynB2xn. Then by Hahn-Banach theorem there isφn ∈E∗ withφn|F φn

andφnφn, for eachn. So

φnχnφn

B1−B_B1₂B2

xn

0, 2.28

hence

lim

n φn

χnφn

B1−B_B1₂B2

xn

0,

φnB2xn B2xn2, φnB2xn.

2.29

Thus, 0∈ MB1−B1/B2B2B2and by Lemma2.1

0∈

MB1_B₂−_BB1₂B22

MB1_B₂− B1B2. 2.30

Therefore,

B1B2 ∈ MB1_B₂. 2.31

FromM_A₁_,B₁ M_A₂_,B₂A1B1 A2B2we can find a normalized sequences

Xn_n⊆BEsuch that

lim

n A1XnB1 A2XnB2A1B1 A2B2. 2.32

SinceA1XnB1 A2XnB2B1†X †

nA†1 B †

2X

†

nA†2, for eachn, then we can find a normalized

φnk ∈E†such that

lim

k,n

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We argue similarly and get

lim

k,n

A†₁φnkA†1, lim_k,nA

†

2φnkA†2. 2.34

Following the same steps as in the previous case we obtainA1A2 ∈ MA†₁_A†

2.

Moreover, if we have RanA†₁⊥B−J RanA†2 − A2/A1A†1 and RanB1⊥B−J

RanB2 −B2/B1B1, it suﬃces to reverse, in the proof of the previous case, the role

ofA†₁intoA†₂andB1intoB2.

For the completeness of the previous theorem we need to prove the following result which is very interesting.

We recall that Phelps10has proved that, for a Banach spaceE,∪{Dx :x ∈E}is dense inE∗; this property is called subreflexivity of the spaceE. Using this fact, Bonsall and Duncan2has proved that for any operatorT ∈BEwe haveWT WT†. The following result generalizes the Bollobas result in the caseMA_B, whereA, B∈BE.

Proposition 2.8. LetEbe a Banach space with smooth dual and letA, B ∈ BEsuch thatBis a

surjective operator. ThenMA†_B†⊆ MA_B.

Proof. Let a ∈ MA†_B†, then there are ψn ∈ DB†ϕn, ϕn_n ∈ SE∗B† such that a

limnψnA†ϕn.

By the subreflexivity ofEthere exist sequencesϕ_n_k_n_k ⊆ E∗ andx_n_k ⊆ Esuch that

ϕnk ∈ Dxnk and ϕnk − Bxnkϕn to 0. It follows that the sequence xn k ⊆ E∗∗ has an

E∗∗_{-weak convergent subsequence}_xn

mnm, that is,

xnm

f−→Ψf, ∀f ∈E∗_,_Ψ_∈_E∗∗_. _2.35

On the one hand, we have

Bxnm2

B†_ϕ

nm− Bxnmϕn

xnm Bxnm B†ϕn

xnm. 2.36

Then

Bxnm2≤B†

ϕnm− Bxnmϕn BxnmB†ϕn. 2.37

Thus

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On the other hand,

xnm B†ϕn

−B†_ϕ

n≤

xnm B†ϕn

−xnm

B†_ϕn m Bxnm

_Bx1_n

m

xnm B†ϕnm

−B†_ϕ

n

x

B†_ϕn₋ 1

Bxnm

B†_ϕn m

Bxnm −B†ϕn

−→0 asm−→ ∞.

2.39

So limmxn mB†ϕn B†ϕnandB†ϕnΨn ∈DB†ϕn. Then by smoothness of the spaceE∗

we getB†ϕ_nΨ_n Ψ_n,for alln. Next,

xnm A†ϕnm

− Bxnm

_B†_ϕ_nψn A†ϕn≤xnm A†ϕnm

−xn m BxnmA†ϕn

Bxnm

xnm A†ϕn

− 1

B†_ϕ_nψn A†ϕn

xnm A†ϕnm−A†ϕn

Bxnmxnm A†ϕn

−ψn A†_ϕn

−→0 asm−→ ∞.

2.40

Then limmxnmA†ϕnm ψnA†ϕnor limmϕnmAxnm ψA†ϕnand therefore

lim

n

lim

m ϕnmAxnm

lim

n ψn A

†_ϕn_a _2.41

which means thata∈ MA_B.

Corollary 2.9. LetEbe a Banach space with smooth dual andA1, A2, B1, B2∈BE.

If M_A₁_,B₁ M_A₂_,B₂ A1B1 A2B2 and RanA†₂⊥_B₋_J RanA†₁ −A1/

A2A†₂withA2being surjective, and RanB2⊥B−JRanB1−B1/B2B2, then

A1A2 ∈ MA1A2, B1B2 ∈ MB1B2. 2.42

Moreover, if RanA†₁⊥_B₋_JRanA†₂ − A2/A1A†₁, A2 is surjective, and RanB1⊥B−J

RanB2−B2/B1B1, then

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Corollary 2.10. LetE be a Banach space with smooth dual and A1, A2, B1, B2 ∈ BEsuch that

A1, A2 are surjective operators. If RanA†_i⊥B−JRanA†j − Aj/AiAi† and RanBi⊥B−J

RanBj−Bj/BiBi, i, j1,2 such thati /jthen the following assertions are equivalent:

1MA1,B1 MA2,B2A1B1 A2B2;

2A1A2 ∈ MA1_A₂∩ MA2_A₁andB1B2 ∈ MB1_B₂∩ MB2_B₁.

As a particular case, we obtain the following.

Corollary 2.11. LetEbe a Banach space with smooth dual andA, Bare surjective operators inBH.

If

Ran B†⊥B−J Ran

A†₋A

BB†

, RanA⊥B−J Ran

B− B

AA

, 2.44

then the following assertions are equivalent:

1AB ∈ MA_B∩ MB_A;

2MA,B MB,A2AB.

3. Hilbert Space Case

LetE Hbe a complex Hilbert space and A ∈ BH. The maximal numerical range ofA

11denoted byW0Ais defined by

{λ∈C:∃x_n,x_n1,such that limAx_n, x_nλ and limAx_nA}, 3.1

and its normalized maximal range, denoted byWNA, is given by

WNA

⎧ ⎨ ⎩

W0

_A

A

ifA /0

0 ifA0.

3.2

The setW0Ais nonempty, closed, convex, and contained in the closure of the numerical

range ofA.

In this section we prove that ifEH, the conditions

A1A2 ∈ MA1_A₂∩ MA2_A₁, B1B2 ∈ MB1_B₂∩ MB2_B₁ 3.3

would imply that

_A∗

2A1A1A2, B2B1∗B1B2,

WNA∗

2A1

∩WNB2B∗₁

/

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Proposition 3.1. LetHbe a complex Hilbert space,A1, A2, B1, B2∈BH.

IfA1A2 ∈ MA1A2∩MA2A1 and B1B2 ∈ MB1B2∩MB2B1, thenA

∗

2A1

A1A2andB2B∗₁B1B2andWNA∗2A1∩WNB2B₁∗/∅.

Proof. IfA10 orA20 andB10 orB20, the result is obvious.

The proof will be done in four steps, we choose one and the others will be proved similarly. Suppose thatA1/0 andA2/0, ifA1A2 ∈ MA1A2, then there exists a sequence

xn_n∈SHA2such that

A1A2limA1xn, A2xn. 3.5

We have|A∗₂A1xn, xn| ≤ A∗₂A1 ≤ A1A2; this yields

limA∗₂A1xnA2∗A1A1A2. 3.6

From3.5and3.6we get

_A∗

2A1A1A2, 1∈W0

A∗

2A1

_A∗

2A1

. 3.7

Suppose now that B1/0 andB2/0, if B1B2 ∈ MB1B2, then there exists a sequence

ynn∈SHB2such that

B1B2limB1yn, B2yn. 3.8

Since limnB1ynB1, then limnB1∗B1yn− B1

2_yn _0.

Suppose thatw_nB1yn/B1, thenynB1∗wn/B1 znsuch that limnzn0.

Hence

B2yn, B1yn

B2

_B∗

1wn

B1

,B1wn

B2B1∗wn, wn

B2zn,B1wn.

3.9

From this, we derive that

limB2B∗1wnB2B∗1B1B2. 3.10

From3.8and3.10we have

_B₂_B∗

1B1B2, 1∈W0

B2B₁∗

_B₂_B∗

1

. 3.11

From3.7and3.11we getA∗₂A1 A1A2andB2B₁∗ B1B2andWNA∗₂A1∩

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Remark 3.2. We remark that in the caseEHwe obtain an implication given by Boumazgour

12.

Acknowledgments

The authors would like to sincerely thank the anonymous referees for their valuable comments which improved the paper. This research was supported by a grant from King Khalid Universityno. KKU S130 33.

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