Generalized derivation modulo the ideal of all compact operators

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GENERALIZED DERIVATION MODULO THE IDEAL

OF ALL COMPACT OPERATORS

SALAH MECHERI and AHMED BACHIR

Received 3 December 1999

We give some results concerning the orthogonality of the range and the kernel of a gener-alized derivation modulo the ideal of all compact operators.

2000 Mathematics Subject Classification: 47B47, 47A30, 47B20, 47B10.

1. Introduction. Letᏸ(Ᏼ)be the algebra of all bounded operators acting on a com-plex Hilbert spaceᏴ. ForAandBinᏸ(Ᏼ), letδA,Bdenote the operator onᏸ(Ᏼ)defined byδA,B(X)=AX−XB. IfA=B, thenδA is called the inner derivation induced byA. In [1, Theorem 1.7], Anderson showed that ifAis normal and commutes withT then, for allX∈ᏸ(Ᏼ),

_{T−(AX−XA)≥ T.} (1.1)

In [4], we generalized this inequality, we showed that if the pair (A,B) has the Putnam-Fuglede’s property (in particular ifAandB are normal operators) andAT=

T B, then for allX∈ᏸ(Ᏼ),

_{T−(AX−XB)≥ T.} (1.2)

The related inequality (1.1) was obtained by Maher [3, Theorem 3.2] who showed that, ifA is normal andAT =T A, whereT ∈Cp, thenT−(AX−XA)p≥ Tp for all

X∈ᏸ(Ᏼ), whereCpis the von Neumann-Schatten class, 1≤p <∞, and·pits norm. Here we show that Maher’s result is also true in the case whereCpis replaced by᏷(Ᏼ), the ideal of all compact operators with·∞its norm. Which allows to generalize these results, we prove that if the pair(A,B)has(PF)᏷(Ᏼ), the Putnam-Fuglede’s property in ᏷(Ᏼ), andAT=T B, whereT∈᏷(Ᏼ), thenT−(AX−XB)∞≥ T∞for allX∈ᏸ(Ᏼ).

2. Normal derivations. In this section, we investigate on the orthogonality of the range and the kernel of a normal derivation modulo the ideal of all compact operators. We recall that the pair(A,B)has the property(PF)᏷(Ᏼ) ifAT =T B, whereT∈᏷(Ᏼ) impliesA∗_{T=T B}∗_{. Before proving this result we need the following lemmas.}

Proof. Letλ1,λ2,...,λnbe eigenvalues of the diagonal operatorN. Then, the op-eratorNcan be written under the following matrix form:

       

λ1 0 ··· 0

0 λ2 ...

..

. . .. 0

0 ··· 0 λn

        . (2.1)

According to the following decomposition ofᏴ:

Ᏼ=

n

i=1

ker N−λj. (2.2)

Let|δij|and |Xij|be the matrix representations ofS andXaccording to the above decomposition ofᏴ. Then

NX−XN= λi−λjXij. (2.3)

SinceS∈ {N}_{(the commutant ofN), we getS}

ij=0 fori≠j. Consequently

NX−XN+S=      

S11 ∗ ∗ ∗ ∗ S22 ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗ Snn

    

. (2.4)

Here∗stands for some entry.

AsδN(X)+S∈᏷(Ᏼ), soS∈᏷(Ᏼ)and the result of Gohberg and Kre˘ın [2] guarantee thatδN(X)+S∞≥ S∞.

Lemma2.2. LetN∈ᏸ(Ᏼ)be a normal operator and letᏴ1=Vectλ∈Cker(N−λ). IfS∈ {N}and there existsX∈ᏸ(Ᏼ)such thatδN(X)+S∈᏷(Ᏼ), thenᏴ1reducesS and the restrictionS|Ᏼ⊥1=0.

Proof. SinceNis a normal operator,Ᏼ1reducesNand the restrictionN|Ᏼ1is a diagonal operator, then the Putnam-Fuglede’s theorem guarantees that S∗ ∈ {N}. Hence,Ᏼ1reducesS. Let

N=

N1 0

0 N2

, S=

S1 0

0 S2

, X=

X11 X12 X21 X22

(2.5)

on Ᏼ=Ᏼ1⊕Ᏼ2, where Ᏼ2 =Ᏼ⊥1. The hypothesis δN(X)+S ∈᏷(Ᏼ)would imply

thatδN2(X22)+S2∈᏷(Ᏼ). The result of Anderson [1] (applied to the Calkin algebra ᏸ(Ᏼ2)\᏷(Ᏼ2)) guarantees that S2∈᏷(Ᏼ). Since the normal operatorN2is without

eigenvalues and the selfadjoint operatorS∗

2S2is compact and belongs to the

Theorem 2.3. LetN∈ᏸ(Ᏼ)be a normal operator,S ∈ {N}_{, andX∈ᏸ(Ᏼ). If}

δN(X)+S∈᏷(Ᏼ), thenS∈᏷(Ᏼ)and

_{δN(X)+S∞≥ S∞.} (2.6)

Proof. SinceδN(X)+S∈᏷(Ᏼ), it follows fromLemma 2.2that

N=

N1 0

0 N2

, S=

S1 0

0 S2

(2.7)

onᏴ=Ᏼ1⊕Ᏼ⊥1, whereᏴ1=Vectλ∈Cker(N−λ). If

X=

X11 X12 X21 X22

(2.8)

onᏴ=Ᏼ1⊕Ᏼ⊥1, then

δN(X)+S=

δN1 X11

+S1 ∗

∗ ∗

. (2.9)

SinceδN(X)+S ∈᏷(Ᏼ), it results that δN1(X11)+S1∈᏷(Ᏼ). As N is a diagonal operator andS1∈ {N1}, it follows fromLemma 2.1thatS1is compact and

_δN 1 X11

+S1_∞≥S1_∞. (2.10)

Consequently,Sis compact and

_{δN(X)+S∞≥δN} 1 X11

+S1_∞≥S1_∞= S∞. (2.11)

Corollary2.4. LetN,M,S∈ᏸ(Ᏼ)such thatNandMare normal operators and NS=SM. IfX∈ᏸ(Ᏼ)such thatδN,M(X)+S∈᏷(Ᏼ), thenS∈᏷(Ᏼ)and

_δN,M(X)+S

∞≥ S∞. (2.12)

Proof. Consider the operatorsL,T, andY defined onᏴ=Ᏼ⊕Ᏼby

L=

N 0

0 M

, S=

0 S

0 0

, Y=

0 X

0 0

, (2.13)

thenLis normal,T∈ {L}_and

δL(Y )+T=

0 δN,M(X)+S

0 0

. (2.14)

ThenTheorem 2.3would imply thatT is compact and

_{δL(Y )+T}

consequently,Sis compact and

_δN,M(X)+S

∞≥ S∞. (2.16)

3. Generalized derivations. In this section, we generalize the above results to a large class of operators. We show that if the pair(A,B)has the property(PF)᏷(Ᏼ), and

AS=SBsuch thatδN,M(X)+S∈᏷(Ᏼ), thenS∈᏷(Ᏼ)and

_δA,B(X)+S

∞≥ S∞, ∀x∈ᏸ(Ᏼ). (3.1)

Before proving this result, we need the following lemma.

Lemma3.1. LetA,B∈ᏸ(Ᏼ). The following statements are equivalent: (1) the pair(A,B)has the property(PF)᏷(Ᏼ);

(2) ifAT =T B, whereT ∈᏷(Ᏼ), then R(T )reduces A, ker(T )⊥ reduces B, and A|R(T )andB|ker(T )⊥ are normal operators.

Proof. (1)⇒(2). Since ᏷(Ᏼ) is a bilateral ideal andT ∈᏷(Ᏼ), thenAT ∈᏷(Ᏼ). Hence, asAT=T Band(A,B)satisfies(PF)᏷(Ᏼ),A∗T=T B∗andR(T ), and ker(T )⊥are reducing subspaces for A and B, respectively. Since A(AT ) = (AT )B implies

A∗_{(AT )=(AT )B}∗ _{by (PF)᏷(Ᏼ), and the identityA}∗_{T = T B}∗ _{implies that A}∗_{AT =}

AA∗_{T, thus we see thatA|}

R(T )is normal. Clearly,(B∗,A∗)satisfies(PF)᏷(Ᏼ)andB∗T∗=

T∗_A∗_{. Therefore, it follows from the above argument thatB}∗_|

R(T∗₎=B|ker(T )⊥is nor-mal.

(2)⇒(1). LetT ∈᏷(Ᏼ)such thatAT =T B. Taking the two decompositions of Ᏼ, Ᏼ1=Ᏼ=R(T )⊕R(T )⊥andᏴ2=Ᏼ=ker(T )⊥⊕kerT. Then we can writeAandBon Ᏼ1intoᏴ2, respectively,

A=

A1 0

0 A2

, B=

B1 0

0 B2

, (3.2)

whereA1andB1are normal operators. Also we can writeTandXonᏴ2intoᏴ1

T=

T1 0

0 0

, X=

X1 X2 X3 X4

. (3.3)

It follows from AT =T B that A1T1=T1B1. Since A1and B1 are normal operators,

then, by applying the Fuglede-Putnam’s theorem, we obtain A∗

1T1= T1B1∗, that is, A∗_{T=T B}∗_.

Theorem3.2. LetA,B∈ᏸ(Ᏼ)satisfying(PF)᏷(Ᏼ) andAS=SB. IfX∈ᏸ(Ᏼ)such thatδA,B(X)+S∈᏷(Ᏼ), thenS∈᏷(Ᏼ)and

_δA,B(X)+S

Proof. Since the pair(A,B)satisfies the property(PF)᏷(Ᏼ), it follows byLemma 3.1 thatR(S)reducesA, ker(S)⊥_{reducesB, andA|}

R(S)andB|ker(S)⊥are normal operators. LetᏴ1=R(S)⊕R(S)⊥andᏴ2=ker(S)⊥⊕kerS. Then

A=

A1 0

0 A2

, B=

B1 0

0 B2

,

S=

S1 0

0 0

, X=

X1 X2 X3 X4

.

(3.5)

It follows from

AS−SB=

A1S1−S1B1 0

0 0

=0 (3.6)

thatA1S1=S1B1and we have

_{S−(AX−XB)}

∞=

S1− A1X1−X1B1

∗

∗ ∗

_∞. (3.7)

SinceA1andB1are two normal operators, then it results fromCorollary 2.4thatS1

is compact and

_{S1− A1X1−X1B1∞≥S1∞,} (3.8)

so

_{S−(AX−XB)∞≥S1− A1X1−X1B1}

∞≥S1_∞= S∞. (3.9)

Corollary3.3. LetA,B∈ᏸ(Ᏼ)satisfying(PF)᏷(Ᏼ)andAS=SB. IfX∈ᏸ(Ᏼ)such thatδA,B(X)+S∈᏷(Ᏼ), thenS∈᏷(Ᏼ)and

S+AX−XB∞≥ S∞ (3.10)

in each of the following cases:

(1) ifA,B∈ᏸ(Ᏼ)such thatAx ≥ x ≥ Bxfor allx∈Ᏼ; (2) ifAis invertible andBsuch thatA−1_{B ≤1.}

Proof. (1) The result of Tong [5, Lemma 1] guarantees that the above condition implies that for all T ∈ker(δA,B|᏷(Ᏼ)), R(T )reducesA, ker(T )⊥ reducesB, and

A|R(T )andB|ker(T )⊥ are unitary operators. Hence, it results fromLemma 3.1that the pair(A,B)has the property(PF)᏷(Ᏼ)and the result holds byTheorem 3.2.

Inequality (3.10) holds in particular ifA=Bis isometric; in other words,Ax = x

for allx∈Ᏼ.

References

[1] J. H. Anderson,On normal derivations, Proc. Amer. Math. Soc.38(1973), 135–140. [2] I. C. Gohberg and M. G. Kre˘ın,Introduction to the Theory of Linear Nonselfadjoint Operators,

Translations of Mathematical Monographs, vol. 18, American Mathematical Society, Rhode Island, 1969.

[3] P. J. Maher,Commutator approximants, Proc. Amer. Math. Soc.115(1992), no. 4, 995– 1000.

[4] S. Mecheri,On minimizingS−(AX−XB)pp, Serdica Math. J.26(2000), no. 2, 119–126. [5] Y. S. Tong,Kernels of generalized derivations, Acta Sci. Math. (Szeged)54(1990), no. 1-2,

159–169.

Salah Mecheri: Department of Mathematics, College of Science, King Saud

Univer-sity, P.O. Box₂₄₅₅, Riyadh₁₁₄₅₁, Saudi Arabia

E-mail address:[email protected]

Ahmed Bachir: University of Mostaganem, Exacte Sciences Institute, BP₂₂₇,₂₇₀₀₀ Mostaganem, Algeria