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volume 6, issue 3, article 90, 2005.

Received 02 April, 2003;

accepted 16 June, 2005.

Communicated by:A. Babenko

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Journal of Inequalities in Pure and Applied Mathematics

ANOTHER VERSION OF ANDERSON’S INEQUALITY IN THE IDEAL OF ALL COMPACT OPERATORS

SALAH MECHERI

King Saud University, College of Science Department of Mathematics

P.O. Pox 2455, Riyah 11451 Saudi Arabia

EMail:mecherisalah@hotmail.com

c

2000Victoria University ISSN (electronic): 1443-5756 041-03

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Another Version of Anderson’s Inequality in the Ideal of all

Compact Operators Salah Mecheri

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J. Ineq. Pure and Appl. Math. 6(3) Art. 90, 2005

http://jipam.vu.edu.au

Abstract

This note studies how certain problems in quantum theory have motivated some recent research in pure Mathematics in matrix and operator theory. The math- ematical key is that of a commutator. We introduce the notion of the pair(A, B) of operators having the Fuglede-Putnam’s property in the ideal of all compact operators. The characterization of this class leads us to generalize some recent results. We also give some applications of these results.

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

Key words: Generalized derivation, Orthogonality, Compact operators.

I would like to thank the referee for his careful reading of the paper. His valuable sug- gestions, critical remarks, and pertinent comments made numerous improvements throughout. This research was supported by the K.S.U. research center project no.

Math/2005/04.

1. Introduction

LetHdenote a separable infinite-dimensional complex Hilbert space. Let L(H)⊃ K(H)⊃Cp ⊃ F(H)

(0 < p < ∞ ) denote, respectively, the class of all bounded linear operators, the class of compact operators, the Schattenp-class, and the class of finite rank operators onH. All operators herein are assumed to be linear and bounded. Let k·k_p,k·k_∞ denote, respectively, theC_p- norm and theK(H)-norm. LetI be a proper bilateral ideal ofL(H). It is well known that ifI 6= {0}, thenK(H)⊃ I ⊃ F(H). ForA, B ∈ L(H) we define the generalized derivation δA,B as follows

δ_A,B(X) = AX−XB

forX ∈ L(H)(so thatδ_A,A=δ_A). In [1, Theorem 1.7], J. Anderson shows that ifAis normal and commutes withT then,

(1.1) kT −(AX−XA)k ≥ kTk,

for all X ∈ L(H). In [11] we generalized this inequality, showing that if the pair (A, B) has the Fuglede-Putnam’s property (in particular if A and B are normal operators) andAT =T B,then for allX ∈ L(H),

kT −(AX−XB)k ≥ kTk.

The related inequality (1.1) was obtained by P.J. Maher [13, Theorem 3.2] showing that ifAis normal andAT =T A,whereT ∈C_p, then

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kT −(AX−XA)k_p ≥ kTk_p

for allX ∈ L(H), whereCp is the von Neumann-Schatten class,

1 ≤ p < ∞ andk·k_p its norm. In [12] we generalized P.J. Maher’s result, proving that if the pair(A, B)has the Fuglede-Putnam’s property(F P)_C_p, then

kT −(AX−XB)k_p ≥ kTk_p

for allX ∈ L(H), and for all T ∈C_p∩kerδ_A,B. In [9] F. Kittaneh shows that if the pair(A, B)has the Fuglede-Putnam’s property inL(H)then

kT −(AX−XB)k_I ≥ kTk_I

for all X ∈ L(H), and for allT ∈ I ∩kerδ_A,B. In order to generalize these results, we prove that if the pair(A, B)has the(F P)K(H)property (the Fuglede- Putnam’s property inK(H)), then

kT −(AX−XB)k_∞≥ kTk_∞

for allX ∈ K(H)and for allT ∈ K(H)∩kerδ_A,B. That is, the zero generalized commutator is the generalized commutator inK(H)ofT.

A.H. Almoadjil [2] shows that if A is normal and for every X ∈ L(H), A²X = XA² andA³X = XA³, thenAX = XA. However F. Kittaneh [7]

generalizes the Almoadjil’s theorem by choosing Aand B^∗ subnormal. There are of course other co-prime pairs of powers ofAandB, such as2and2n+ 1 or3and2n+ 1(with3and2n+ 1co-prime), for which a similar result can be proved. Notice here that for such co-prime powers ofAandB, the hypothesis

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that the pair (A, B) has the(F P)K(H) property implies thatδ_A,B^m (X) = 0for some integer m >1,and the conclusionX ∈kerδA,Bis a consequence of the following general result: Let δ_A,B^m denote an m−times application of δ_A,B. If the pair (A, B)has the(F P)K(H)property and δ_A,B^m (X) = 0 for some integer m >1, thenδA,B(X) = 0.

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2. Orthogonality

We begin by the following definition of the orthogonality in the sense of G.

Birkhoff [3] which generalizes the idea of orthogonality in Hilbert space.

Definition 2.1. Let Cbe the field of complex numbers and letE be a normed linear space. Letx, y ∈E. Ifkx−λyk ≥ kλykfor allλ∈C, thenxis said to be orthogonal toy. LetF andGbe two subspaces inE. Ifkx+yk ≥ kyk, for allx∈F and for ally∈G, thenF is said to be orthogonal toG.

Definition 2.2. LetA, B ∈ L(H). We say that the pair(A, B)satisfies(F P)K(H), ifAC =CB whereC ∈ K(H)impliesA^∗C =CB^∗.

Theorem 2.1. LetA, B ∈ L(H). IfAandBare normal operators, then kS−(AX−XB)k_∞≥ kSk_∞

for allX ∈ L(H)and for allS ∈kerδ_A,B∩ K(H).

Proof. LetS =U|S|be the polar decomposition ofS, whereU is an isometry such thatkerU = ker|S|. Since

kU^∗Sk_∞≤ kU^∗k_∞kSk_∞ =kSk_∞ for allS ∈ K(H),

kS−(AX−XB)k_∞≥sup

n

|(U^∗[S−(AX−XB)]ϕ_n, ϕ_n)|

(2.1)

= sup

n

([|S| −U^∗(AX−XB)]ϕn, ϕn)

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for any orthonormal basis{ϕ_n}_n≥1ofH. SinceAS =SBandA, B are normal operators, then it follows from the Fuglede-Putnam’s theorem thatS^∗A=BS^∗; consequentlyS^∗AS =BS^∗SorS^∗SB = BS^∗S, i.e,B|S|=|S|B.Since|S|

is a compact normal operator and commutes with B, there exists an orthonormal basis {fk} ∪ {gm} ofH such that {fk} consists of common eigenvectors of B and |S|, and {g_m} is an orthonormal basis of ker|S|. Since {f_k} is an orthonormal basis of the normal operatorB, then there exists a scalarα_k such thatfk =αkfkandB^∗fk=αkfk; consequently

hU^∗(AX−XB)f_k,|S|f_ki=hS^∗(AX−XB)f_k, f_ki

=h(B(S^∗X)−(S^∗X)B)f_k, f_ki= 0.

That is, hU^∗(AX −XB)f_k, f_ki= 0. In (2.1) take{ϕ_n} = {f_k} ∪ {g_m}as an orthonormal basis ofH. Then

kS−(AX−XB)k_∞ ≥sup

n

([|S| −U^∗(AX−XB)]ϕ_n, ϕ_n)

= sup

k,m

[|S|f_k, f_k) + (U^∗(AX−XB)g_m, g_m)]

≥sup

k

(|S|f_k, f_k)

=k|S|k=kSk_∞.

Theorem 2.2. Let A, B ∈ L(H). If the pair (A, B) satisfies the (F P)_K(H₎ property, then

(2.2) kδA,B(X) +Sk_∞≥ kSk_∞,

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for allX ∈ K(H), and for allS ∈ K(H)∩ker(δ_A,B).In particular we have (2.3) R(δ_A,B |K(H))∩ker(δ_A,B |K(H)) = {0},

whereR(δ_A,B)andker(δ_A,B)denote the range and the kernel ofδ_A,B.

Proof. It is well known that if the pair(A, B)satisfies the(F P)K(H)property, thenR(S)reducesA, ker^⊥S reducesB andA |_R(S), B |_ker^⊥_S are normal operators. Letting S₀ : ker^⊥S → R(S) be the quasi-affinity defined by setting S0x=Sxfor eachx∈ker^⊥S,then it results thatδA1,B1(S0) =δA^∗₁,B^∗₁(S0) = 0.

Let A = A₁⊕A₂,with respect to H = R(S)⊕R(S)^⊥, B = B₁⊕B₂, with respect to H = ker(S)^⊥⊕ kerS and X : R(S)⊕R(S)^⊥ → ker(S)^⊥⊕kerS have the matrix representation

X =

X1 X2

X₃ X₄

.

Then we have

kS−(AX −XB)k_∞ =

S1−(A1X1−X1B1) ∗

∗ ∗

∞

.

The result of I.C. Gohberg and M.G. Krein [6] guarantees that kS−(AX−XB)k_∞ ≥ kS₁−(A₁X₁−X₁B₁)k_∞.

SinceA₁andB₁ are two normal operators, it results from Theorem2.2that kS1−(A1X1−X1B1)k_∞≥ kS1k_∞=kSk_∞

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and

kS−(AX−XB)k_∞ ≥ kS₁−(A₁X₁−X₁B₁)k_∞≥ kS₁k_∞ =kSk_∞.

We can ask “Is the sufficient condition in Theorem2.2necessary?”

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3. Examples and Applications

The related topic of approximation by commutators AX−XA or by generalized commutatorAX−XB, which has attracted much interest, has its roots in quantum theory. The Heinsnberg Uncertainly principle may be mathematically formulated as saying that there exists a pairA, X of linear transformations and a non-zero scalarαfor which

(3.1) AX−XA =αI.

Clearly, (3.1) cannot hold for square matricesAandX and for bounded linear operatorsAandX. This prompts the question:

How close canAX −XAbe the identity?

Williams [17] proved that ifAis normal, then, for allXinB(H),

(3.2) ||I −(AX −XA)|| ≥ ||I||.

Mecheri [14] generalized Williams inequality (3.2): he proved that ifA, B are normal, then for allX ∈B(H)

(3.3) ||I−(AX−XB)|| ≥ ||I||.

Anderson [1] generalized Williams inequality (3.2): he proved that ifAis normal and commutes withBthen, for allX ∈B(H)

(3.4) ||B −(AX −XA)|| ≥ ||B||.

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Maher [13] obtained theC_p variants of Anderson’s result. Mecheri [14] studied approximation by generalized commutators AX −XC: he showed that the following inequality holds

(3.5) ||B−(AX−XC)||_p ≥ ||B||_p,

for all X ∈ C_p if and only ifB ∈ kerδ_A,B. In Theorem2.2 we obtained the K(H)of Maher and Mecheri’s results.

In the previous inequality (3.5) the zero generalized commutator is a generalized commutator approximant inC_P ofB.

Now we are ready to give some operators for which the inequality (2.2) holds.

Corollary 3.1. Let A, B ∈ L(H). Then the pair (A, B) has the (F P)_K(H₎ property in each of the following cases:

(1) IfA, B ∈ L(H)such thatkAxk ≥ kxk ≥ kBxkfor allx∈H.

(2) IfAis invertible andB such thatkA⁻¹k kBk ≤1.

(3) IfA=B is a cyclic subnormal operator.

Proof. The result of Y. Tong [16, Lemma 1] guarantees that the above condition implies that for all T ∈ ker(δ_A,B | K(H)), R(T)reducesA, ker(T)^⊥ reduces B, and A |_R(T₎ and B |_ker(T₎^⊥ are unitary operators. Hence it results from Theorem2.2that the pair(A, B)has the property(F P)K(H)and the result holds by the above theorem. The above inequality holds in particular if A = B is isometric, in other wordskAxk=kxkfor allx∈H.

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(2) In this case it suffices to take A₁ = kBk⁻¹A and B₁ = kBk⁻¹B, then kA₁xk ≥ kxk ≥ kB₁xkand the result holds by (1) for allx∈H.

(3) Since T commutes with A, it follows that T is subnormal [18]. But any compact subnormal operator is normal. Hence T is normal. Now AT = T A impliesA^∗T =T A^∗,i.e, the pair(A, A)has the(F P)K(H)property.

Theorem 3.2. Let A, B ∈ L(H)such that the pairs (A, A) and (B, B) have the(F P)K(H)property. Ifσ(A)∩σ(B) =φ, then

kT −δ_A⊕B,A⊕B(X)k_∞≥ kTk_∞ for allX ∈ K(H), and for allT ∈ K(H)∩ker(δ_A,B).

Proof. It suffices to show that the pair(A⊕B, A⊕B)has the(F P)K(H)property.

Let

T =

T1 T2

T₃ T₄

be in K(H ⊕H).If(A⊕B)T = T(A⊕B), thenAT₁ = T₁A, BT₄ = T₄B, AT₂ = T₂B and BT₃ = T₃A. Since σ(A)∩ σ(B) = φ, then δ_A,B, δ_B,A are invertible [12]. Consequently T₂ = T₃ = 0and since (A, A)and(B, B)have the (F P)_K(H₎ property,AT₁^∗ = T₁^∗AandBT₄^∗ = T₄^∗B, that is,(A⊕B)T^∗ = T^∗(A⊕B).

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4. On the Commutant of A and its Powers

In this section we will be interested on the investigation of the relation between the commutant of a bounded linear operatorAand its powers.

Lemma 4.1. LetA, B ∈ L(H). Then

R(δ_A,B)∩kerδ_A,B ={0} ⇔kerδ_A,B^m = kerδ_A,B, for allm ≥1.

Proof. Suppose thatR(δ_A,B)∩kerδ_A,B ={0}.It suffices to prove that kerδ²_A,B⊂kerδ_A,B.

IfX ∈kerδ²_A,B,thenδA,B(X)∈R(δA,B)∩kerδA,B={0},i.e.X ∈kerδA,B. Conversely ifY ∈R(δ_A,B)∩kerδ_A,B, thenY =δ_A,B(X)for someX ∈ L(H) and δ_A,B(Y) = 0.Consequently we have δ_A,B² (X) = 0,i.e. X ∈ kerδ²_A,B = kerδA,B. Then we obtainδA,B(X) = 0,i.e.Y = 0.

Lemma 4.2. IfR(δA,B)∩kerδA,B ={0}, then

kerδ_A,B =

∞

\

i=2

kerδ_Aⁱ_,Bⁱ.

Proof. Note thatkerδ_A,B ⊂ T∞

i=2kerδ_Aⁱ_,Bⁱ.Hence it suffices to prove the op- posite inclusion. IfX ∈ T∞

i=2kerδ_Aⁱ_,Bⁱ,thenA²X =XB²andA³X =XB³. HenceA²XB =XB³ andAXB² =A³X.LetC =AX−XB.Then,

A²C =A³X−A²XB =XB³−XB³ = 0;

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CB² =AXB²−XB³ =A³X−A³X = 0;

ACB =A²XB −AXB² =XB³−XB³ = 0;

hence

(4.1) A(AC−CB) =A²C−ACB = 0;

(4.2) (AC−CB)B =ACB −CB² = 0.

Thus (4.1) and (4.2) imply that

AC−CB ∈R(δ_A,B)∩kerδ_A,B ={0}, from which it results thatAC =CB. Hence

C ∈R(δ_A,B)∩kerδ_A,B, that is,C = 0and thusAX =XB,i.e,X ∈kerδA,B. Theorem 4.3. If(A, B)has the(F P)_K(H₎property, then

kerδ_A,B^m = kerδ_A,B =

∞

\

i=2

kerδ_Aⁱ_,Bⁱ, m ≥1.

In particular if A²X = XB² and A³X = XB³ for some X ∈ K(H), then AX =XB.

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Proof. This is an immediate consequence of Lemma4.1, Lemma 4.2and The- orem2.2.

Remark 1. The above theorem generalizes the results of F. Kittaneh [9] and Almoadjil [2]. In [8] F. Kittaneh shows that if the pair(A, B)has the(F P)L(H)

property, then for allT ∈ker(δA,B |I)and for allX ∈ I, kδ_A,B(X) +Sk_I ≥ kSk_I.

In Theorem 2.2we show that it suffices that the pair(A, B)has the(F P)K(H)

property for which R(δ_A,B |_K(H))is orthogonal to ker(δ_A,B |_K(H)).The results of this paper are also true in the case whereK(H)is replaced by a two sided ideal of L(H). Hence Theorem2.2 generalizes the results of F. Kittaneh [8], [9] and of S. Mecheri [12].

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References

[1] J. ANDERSON, On normal derivation, Proc. Amer. Math. Soc., 38 (1973), 135–140.

[2] A.H. AlMOADJIL, On the commutant of relatively prime powers in Ba- nach algebra, Proc. Amer. Math. Soc., 57 (1976), 243–249.

[3] G. BIRKHOFF, Orthogonality in linear metric space spaces, Duke Math.J., 1 (1935), 169–172.

[4] B.P. DUGGAL, On generalized Putnam-Fuglede theorem, Monatsh.

Math., 107 (1989), 309–332.

[5] B.P. DUGGAL, A remark on normal derivations of Hilbert-Schmidt type, Monatsh. Math., 112 (1991), 265–270.

[6] I.C. GOHBERG AND M.G. KREIN, Introduction to the range of linear nonselfadjoint operators, Trans. Math. Monographs, 18, Amer. Math. Soc, Providence, RI, 1969.

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[8] F. KITTANEH, Operators that are orthogonal to the range of a derivation, Jour. Math. Anal. Appl., 203 (1997), 868–873.

[9] F. KITTANEH, On normal derivations in norm ideal, Proc. Amer. Math.

Soc., 123 (1995), 1779–1785.

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[10] F. KITTANEH, On a generalized Fuglede-Putnam theorem of Hilbert Schmidt type, Proc. Amer. Math. Soc., (1983), 293–298.

[11] S. MECHERI, On minimizingkT −(AX−XB)k^p_p, Serdica. Math. J., 26 (2000), 119–126.

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[16] Y. TONG, Kernels of generalized derivations, Acta. Sci. Math., 102 (1990), 159–169.

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