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volume 7, issue 2, article 77, 2006.

Received 01 September, 2005;

accepted 27 January, 2006.

Communicated by:A.G. Babenko

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Journal of Inequalities in Pure and Applied Mathematics

GÂTEAUX DERIVATIVE AND ORTHOGONALITY IN C_p-CLASSES

SALAH MECHERI

King Saud University, College of Sciences Department of Mathematics

P.O. Box 2455, Riyah 11451 Saudi Arabia

EMail:mecherisalah@hotmail.com

2000c Victoria University ISSN (electronic): 1443-5756 259-05

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Gâteaux Derivative and Orthogonality inCp-classes

Salah Mecheri

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J. Ineq. Pure and Appl. Math. 7(2) Art. 77, 2006

http://jipam.vu.edu.au

Abstract

The general problem in this paper is minimizing theC_p−norm of suitable affine mappings from B(H) toCp, using convex and differential analysis (Gateaux derivative) as well as input from operator theory. The mappings considered generalize the so-called elementary operators and in particular the generalized derivations, which are of great interest by themselves. The main results obtained characterize global minima in terms of (Banach space) orthogonality, and constitute an interesting combination of infinite-dimensional differential analysis, operator theory and duality. Note that the results obtained generalize all results in the literature concerning operator which are orthogonal to the range of a derivation and the techniques used have not been done by other authors.

2000 Mathematics Subject Classification: Primary 47B47, 47A30, 47B20; Sec- ondary 47B10.

Key words: Elementary operators, Schatten p-classes, orthogonality, Gateaux derivative.

This work was supported by the College of Science Research Center Project No.

Math/2006/23.

1. Introduction

LetE be a complex Banach space. We first define orthogonality inE. We say thatb∈E is orthogonal toa∈Eif for all complexλthere holds

(1.1) ka+λbk ≥ kak.

This definition has a natural geometric interpretation. Namely,b⊥aif and only if the complex line{a+λb|λ∈C}is disjoint with the open ballK(0,kak), i.e., iff this complex line is a tangent one. Note that if b is orthogonal to a, then a need not be orthogonal to b. If E is a Hilbert space, then from (1.1) follows ha, bi = 0, i.e., orthogonality in the usual sense. Next we define the von Neumann-Schatten classesC_p (1≤p <∞). LetB(H)denote the algebra of all bounded linear operators on a complex separable and infinite dimensional Hilbert space H and let T ∈ B(H) be compact, and let s₁(T) ≥ s₂(T) ≥

· · · ≥ 0denote the singular values ofT, i.e., the eigenvalues of|T| = (T^∗T)¹² arranged in their decreasing order. The operator T is said to belong to the Schattenp-classesC_p if

kTk_p =

" _∞ X

i=1

si(T)^p

#¹_p

= [tr|T|^p]¹^p <∞, 1≤p < ∞,

where tr denotes the trace functional. Hence C₁ is the trace class, C₂ is the Hilbert-Schmidt class, and C_∞ corresponds to the class of compact operators with

kTk_∞=s₁(T) = sup

kfk=1

kT fk

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denoting the usual operator norm. For the general theory of the Schatten p- classes the reader is referred to [16]. Recall (see [16]) that the normk·kof the B−spaceV is said to be Gâteaux differentiable at non-zero elementsx ∈V if there exists a unique support functional (in the dual spaceV^∗) such thatkD_xk= 1andDx(x) = kxk, satisfying

lim

R3t→0

kx+tyk − kxk

t = ReD_x(y),

for ally ∈V. HereRdenotes the set of all reals and Redenotes the real part.

The Gâteaux differentiability of the norm atximplies thatxis a smooth point of a sphere of radiuskxk.

It is well known (see [6] and the references therein) that for 1 < p < ∞, Cp is a uniformly convex Banach space. Therefore every non-zeroT ∈Cp is a smooth point and in this case the support functional ofT is given by

(1.2) DT(X) = tr

"

|T|^p−1U X^∗ kTk^p−1_p

# ,

for all X ∈ C_p, where T = U|T| is the polar decomposition of T. The first result concerning the orthogonality in a Banach space was given by Anderson [1] showing that ifAis a normal operator on a Hilbert spaceH,thenAS =SA implies that for any bounded linear operatorX there holds

(1.3) kS+AX−XAk ≥ kSk.

This means that the range of the derivation δ_A : B(H) → B(H) defined by δA(X) = AX−XAis orthogonal to its kernel. This result has been generalized

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in two directions: by extending the class of elementary mappings

EA,B:B(H)→B(H); EA,B(X) =

n

X

i=1

AiXBi−X

and

∼

E_A,B :B(H)→B(H);

∼

E_A,B(X) =

n

X

i=1

A_iXB_i,

where(A₁, A₂, . . . , A_n)and(B₁, B₂, . . . , B_n)aren−tuples of bounded operators onH, and by extending the inequality (1.3) toCp-classes with1< p <∞ see [3], [6], [9]. The Gâteaux derivative concept was used in [3,5,6,7,15] and [8], in order to characterize those operators which are orthogonal to the range of a derivation. The main purpose of this note is to characterize the global minimum of the map

X 7→ kS+φ(X)k_C

p, φis a linear map inB(H),

in C_p by using the Gateaux derivative. These results are then applied to characterize the operatorsS ∈ C_p which are orthogonal to the range of elementary operators. It is very interesting to point out that our Theorem2.3and its Corol- lary2.6generalize Theorem 1 in [6] , Lemma 2 in [3] and Theorem 2.1 in [18].

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2. Main Results

Let φ : B(H) → B(H) be a linear map, that is, φ(αX +βY) = αφ(X) + βφ(Y), for allα, β ∈ Cand allX, Y ∈ B(H), and letS ∈ Cp (1< p < ∞).

Put

U ={X ∈B(H) :φ(X)∈C_p}.

Letψ :U →C_p be defined by

ψ(X) =S+φ(X).

Define the functionFψ :U → R⁺byFψ(X) =kψ(X)k_C

p. Now we are ready to prove our first result in C_p-classes(1 < p < ∞). It gives a necessary and sufficient optimality condition for minimizingF_ψ.

LetX be a Banach space,φa linear mapX →X, andψ(x) = φ(x) +sfor some elements∈X. Use the notation

D_x(y) = lim

t→0⁺

1

t(kx+tyk − kxk).

It is obvious that D_x is sub-additive andD_x(y)≤ kyk, alsoD_x(x) = kxkand D_x(−x) =−kxk.

Theorem 2.1. The map F_ψ = kψ(x)k has a global minimum atx ∈ X if and only if

(2.1) D_ψ(x)(φ(y))≥0, ∀y∈X.

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Proof. Necessity is immediate from ψ(x) +tφ(y) = ψ(x+ty). Sufficency:

assume the stated condition and choosey. Note thatφ(y−x) =ψ(y)−ψ(x).

For brevity we letD_ψ(x)=L. Then kψ(x)k=−L(−ψ(x))

≤ −L(−ψ(x)) +L(ψ(y)−ψ(x)) by hypothesis

≤L(ψ(y)) by sub-additivity

≤ kψ(y)k.

Theorem 2.2 ([7]). LetX, Y ∈Cp. Then, there holds D_X(Y) = pRe

tr(|X|^p−1U^∗Y) ,

whereX =U|X|is the polar decomposition ofX.

The following corollary establishes a characterization of the Gateaux derivative of the norm inCp-classes(1< p < ∞). Now we are going to characterize the global minimum ofF_ψ onC_p (1< p <∞), whenφis a linear map satisfying the following useful condition:

(2.2) tr(Xφ(Y)) =tr(φ^∗(X)Y), ∀X, Y ∈C_p

whereφ^∗ is an appropriate conjugate of the linear mapφ. We state some exam- ples ofφandφ^∗which satisfy the above condition (2.2).

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1. The elementary operator

∼

E_A,B :I → I defined by

∼

E_A,B(X) =

n

X

i=1

A_iXB_i,

whereAi, Bi ∈B(H) (1 ≤i ≤ n)andI is a separable ideal of compact operators inB(H)associated with some unitarily invariant norm. It is easy to show that the conjugate operatorE_A,B^∗ :I^∗ → I^∗ ofE_A,B has the form

∼

E

∗

A,B(X) =

n

X

i=1

B_iXA_i,

and that the operators

∼

EA,B and

∼

E

∗

A,Bsatisfy the condition (2.2).

2. Using the previous example we can check that the conjugate operator E_A,B^∗ :I^∗ → I^∗ of the elementary operator E_A,B defined byE_A,B(X) = Pn

i=1A_iXB_i−X,has the form E_A,B^∗ (X) =

n

X

i=1

B_iXA_i −X,

and that the operatorsE_A,B andE_A,B^∗ satisfy the condition (2.2).

Now, we are in position to prove the following theorem.

Theorem 2.3. LetV ∈C_p, and letψ(V)have the polar decompositionψ(V) = U|ψ(V)|. ThenF_ψ has a global minimum onC_patV if and only if|ψ(V)|U^∗ ∈ kerφ^∗.

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Proof. Assume thatF_ψ has a global minimum onC_p atV. Then

(2.3) D_ψ(V₎(φ(Y))≥0,

for allY ∈C_p.That is, pRe

tr(|ψ(V)|^p−1U^∗φ(Y)) ≥0, ∀Y ∈C_p.

This implies that

(2.4) Re{tr(|ψ(V)|^p−1U^∗φ(Y))} ≥0, ∀Y ∈C_p.

Let f ⊗g, be the rank one operator defined by x 7→ hx, fig where f, g are arbitrary vectors in the Hilbert space H. Take Y = f ⊗g, since the map φ satisfies (2.2) one has

tr(|ψ(V)|^p−1U^∗φ(Y)) = tr(φ^∗(|ψ(V)|^p−1U^∗)Y).

Then (2.4) is equivalent toRe{tr(φ^∗(|ψ(V)|^p−1U^∗)Y)} ≥0,for allY ∈Cp,or equivalently

Re

φ^∗(|ψ(V)|^p−1U^∗)g, f

≥0, ∀f, g∈H.

If we choosef =gsuch thatkfk= 1, we get

(2.5) Re

φ^∗(|ψ(V)|^p−1U^∗)f, f

≥0.

Note that the set {

φ^∗(|ψ(V)|^p−1U^∗)f, f

: kfk= 1}

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is the numerical range of φ^∗(|ψ(V)|^p−1U^∗) on U which is a convex set and its closure is a closed convex set. By (2.5) it must contain one value of positive real part, under all rotation around the origin, it must contain the origin, and we get a vector f ∈ H such that hφ^∗(|ψ(V)|^p−1U^∗)f, fi < where is positive. Since is arbitrary, we get hφ^∗(|ψ(V)|^p−1U^∗)f, fi = 0. Thus φ^∗(|ψ(V)|^p−1U^∗) = 0,i.e.,|ψ(V)|^p−1U^∗ ∈kerφ^∗.

Conversely, ifψ(V)|^p−1U^∗ ∈kerφ^∗, then|ψ(V)|^p−1U^∗ ∈kerφ^∗. It is easily seen (using the same arguments above) that

Re

tr(|ψ(V)|^p−1U^∗φ(Y)) ≥0, ∀Y ∈Cp.

By this we get (2.3).

We state our first corollary of Theorem 2.3. Let φ = δ_A,B, where δ_A,B : B(H)→B(H)is the generalized derivation defined byδA,B(X) =AX−XB.

Corollary 2.4. LetV ∈C_p, and letψ(V)have the polar decompositionψ(V) = U|ψ(V)|. ThenF_ψhas a global minimum onC_patV, if and only if|ψ(V)|^p−1U^∗

∈kerδ_B,A.

Proof. It is a direct consequence of Theorem2.3.

This result may be reformulated in the following form where the global minimum V does not appear. It characterizes the operators S in C_p which are orthogonal to the range of the derivationδ_A,B.

Theorem 2.5. LetS ∈C_p,and letψ(S)have the polar decompositionψ(S) = U|ψ(S)|. Then

kψ(X)k_C_p ≥ kψ(S)k_C_p,

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for allX ∈C_pif and only if|ψ(S)|^p−1U^∗ ∈kerδ_B,A. As a corollary of this theorem we have

Corollary 2.6. LetS ∈Cp∩kerδA,Bhave the polar decompositionS =U|S|.

Then the two following assertions are equivalent:

1.

kS+ (AX−XB)k_C

p ≥ kSk_C

p, for allX ∈C_p. 2. |S|^p−1U^∗ ∈kerδ_B,A.

Remark 1. We point out that, thanks to our general results given previously with more general linear mapsφ, Theorem2.5and its Corollary2.6are true for the nuclear operator∆_A,B(X) = AXB−Xand other more general classes of operators thanδ_A,B such as the elementary operatorsE_A,B(X)and

∼

E_A,B(X).

The above corollary generalizes Theorem 1 in [6]and Lemma 2 in [3].

Now by using Theorem 2.5 , Corollary 2.6, Remark 1 and the following Lemma 2.7 and Lemma 2.9 we obtain some interesting results see also ([3], [13]). LetS =U|S|be the polar decomposition ofS.

Lemma 2.7. LetA, B ∈B(H)andT ∈C_psuch thatkerδ_A,B(T)⊆kerδ_A,B^∗ (T).

If A|S|^p−1U^∗ = |S|^p−1U^∗B, where p > 1 and S = U|S| is the polar decomposition ofS,thenA|S|U^∗ =|S|U^∗B.

Proof. IfT =|S|^p−1, then

(2.6) AT U^∗ =T U^∗B.

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We prove that

(2.7) ATⁿU^∗ =TⁿU^∗B,

for alln ≥1.IfS =U|S|, then

kerU = ker|S|= ker|S|^p−1 = kerT and

(kerU)^⊥= (kerT)^⊥=R(T).

This shows that the projection U^∗U onto (kerT)^⊥ satifies U^∗U T = T and T U^∗U T = T². By taking the adjoints of (2.6) and since kerδ_A,B(T) ⊆ kerδ^∗_A,B(T), we getBU T =U T Aand

AT² =AT U^∗U T =T U^∗BU T =T U^∗U T A=T²A.

SinceAcommutes with the positive operatorT²,Acommutes with its square roots, that is,

(2.8) AT =T A

By (2.6) and (2.8) we obtain (2.7). Letf(t)be the map defined onσ(T)⊂R⁺ byf(t) =t^p−1¹ ; 1 < p <∞.Sincef is the uniform limit of a sequence(P_i)of polynomials without constant term (since f(0) = 0), it follows from (2.8) that AP_i(T)U^∗ =P_i(T)U^∗B.ThereforeAT^p−1¹ U^∗ =U^∗T^p−1¹ B.

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Theorem 2.8. LetA, B be operators inB(H)such thatkerδ_A,B ⊆kerδ_A^∗_,B^∗. ThenT ∈kerδA,B∩Cp,if and only if

kT +δ_A,B(X)k_p ≥ kTk_p, for allX ∈C_p.

Proof. IfT ∈ker ∆_A,B then by applying Theorem 3.4 in [9] it follows that kT +δ_A,B(X)k_p ≥ kTk_p,

for allX ∈C_p.Conversely, if

kT +δ_A,B(X)k_p ≥ kTk_p, for allX ∈Cp,then from Corollary2.6

A|T|U^∗ =|S|U^∗B.

Since kerδ_A,B ⊆ kerδ_A^∗_,B^∗, B^∗|T|^p−1U^∗ = |T|^p−1U^∗A^∗.By taking adjoints we get AU|T|^p−1 = U|T|^p−1B. From Lemma 2.7 it follows that AU|T| = U|T|B.i.e.,T ∈kerδA,B.

Note that the above theorem still holds if we consider∆_A,B instead ofδ_A,B. Let A = (A₁, A₂, . . . , A_n), B = (B₁, B₂, . . . , B_n) be n−tuples of operators in B(H).In the following Theorem 2.11 we will characterizeT ∈ C_p for 1 < p < ∞, which are orthogonal to R(E_A,B | C_p)(the range ofE_A,B | C_p) for a general pair of operators A, B. For this let S = U|S| be the polar decomposition of S. We start by the following lemma or the case where E_C = PC_iXC_i −X which will be used in the proof of Theorem2.11.

LetS =U|S|be the polar decomposition ofS.

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Lemma 2.9. Let C = (C₁, C₂, . . . , C_n)be ann−tuple of operators in B(H) such thatPn

i=1CiC_i^∗ ≤1, Pn

i=1C_i^∗Ci ≤1andkerEC ⊆kerEC^∗. If

n

X

i=1

CiU|S|^p−1Ci =U|S|^p−1,

wherep >1,then

n

X

i=1

C_iU|S|C_i =U|S|.

Proof. IfT =|S|^p−1, then

(2.9)

n

X

i=1

C_iU T C_i =U T.

We prove that

(2.10)

n

X

i=1

C_iU TⁿC_i =U Tⁿ,

It is known that ifPn

i=1C_iC_i^∗ ≤ 1, Pn

i=1C_i^∗C_i ≤ 1andkerE_c ⊆ kerE_c^∗ that the eigenspaces corresponding to distinct non-zero eigenvalues of the compact positive operator|S|²reduces eachC_i(see [4], Theorem 8), ([18], Lemma 2.3)).

In particular, |S| commutes with C_i for all1 ≤ i ≤ n. This implies also that

|S|^p−1 = T commutes with eachCi for all 1 ≤ i ≤ n. HenceCi|T| = |T|Ci

andC_iT² =T²C_i.

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SinceC_icommutes with the positive operatorT², thenC_icommutes with its square roots, that is,

(2.11) C_iT =T C_i.

By the same arguments used in the proof of Lemma2.7the proof of this lemma can be completed.

Theorem 2.10. LetC = (C₁, C₂, . . . , C_n)be ann−tuple of operators inB(H) such that Pn

i=1C_iC_i^∗ ≤ 1, Pn

i=1C_i^∗C_i ≤ 1 andkerE_C ⊆ kerE_C^∗ thenS ∈ kerE_C∩C_p (1< p <∞), if and only if,

kS+E_C(X)k_p ≥ kSk_p,

for allX ∈Cp.

Proof. IfS∈kerE_C then from ([18], Theorem 2.4) it follows that kS+E_C(X)k_p ≥ kSk_p,

for allX ∈C_p.Conversely, if

kS+EC(X)k_p ≥ kSk_p,

for all X ∈ C_p. then from Corollary 2.6 applied for the elementary operator E(X), we get

n

X

i=1

C_i|S|^p−1U^∗C_i =|S|^p−1U^∗.

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SincekerE_C ⊆kerE_C^∗,

n

X

i=1

C_i^∗|S|^p−1U^∗C_i^∗ =|S|^p−1U^∗.

Taking the adjoint we getPn

i=1C_iU|S|^p−1C_i =U|S|^p−1and from Lemma2.9

it follows that _n

X

i=1

C_iU|S|C_i =U|S|,

i.e.,S ∈kerE_C.

Theorem 2.11. LetA= (A₁, A₂, . . . , A_n), B= (B₁, B₂, . . . , B_n)ben−tuples of operators inB(H)such thatPn

i=1A_iA^∗_i ≤1,Pn

i=1A^∗_iA_i ≤1,Pn

i=1B_iB_i^∗ ≤ 1,Pn

i=1B_i^∗B_i ≤1andkerE_A,B ⊆kerE_A^∗_,B^∗. ThenS ∈kerE_A,B∩C_p,if and only if,

kS+E_A,B(X)k_p ≥ kSk_p

for allX ∈C_p.

Proof. It suffices to take the Hilbert spaceH⊕H, and operators

Ci =

A_i 0 0 B_i

, S =

0 T 0 0

, X =

0 X 0 0

and apply Theorem2.11.

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