2000 Mathematics Subject Classification: 47A10, 47A12, 47A30, 47B10, 47B20, 47B37, 47B47, 47D50.We compare the norm of a generalized derivation on a Hilbert space with the norm of its restrictions to Schatten norm ideals

Barraa, Mohamed

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Serdica Math. J. 37 (2011), 107–112GENERALIZED DERIVATIONS AND NORM EQUALITYIN NORMED IDEALSMohamed BarraaCommunicated by I. G. TodorovAbstract. We compare the norm of a generalized derivation on a Hilbertspace with the norm of its restrictions to Schatten norm ideals.Introduction. Let H be a complex Hilbert space and let B(H) denotethe algebra of all bounded linear operators on H. For two bounded operatorsA, B ∈ B(H),the left and right multiplications LA, RB ∈ B(H) are definedby LA(X) = AX and RB(X) = XB respectively. The generalized derivationinduced by A and B is the operatorδA,B : B(H)→ B(H), X 7→ (LA −RB)(X) = AX −XB.The bimultiplication MA,B is the operator defined byMA,B(X) = (RA ◦ LB)(X) = AXB.2010 Mathematics Subject Classification: 47A10, 47A12, 47A30, 47B10, 47B20, 47B37,47B47, 47D50.Key words: Generalized derivation, norm, norm ideal, S-universasl operator, numericalrange, spectrum, quasi-nilpotent operator, hyponormal operator.108 M. BarraaLet (J, ‖.‖J ) be a norm ideal in B(H) in the sense of [13] and let A,B ∈B(H). If X ∈ J , then ‖AX−XB‖J = ‖(A−λ)X−X(B−λ)‖J ≤ (‖A−λ‖+‖B−λ‖)‖X‖J for all λ ∈ C. Hence ‖AX −XB‖J ≤ infλ∈C(‖A−λ‖+ ‖B−λ‖)‖X‖J .Since(1) ‖δA,B‖ = infλ∈C(‖A− λ‖+ ‖B − λ‖)see [15], we conclude that ‖AX − XB‖J ≤ ‖δA,B‖‖X‖J . Thus the restrictionδJ,A,B of δA,B to J defines a bounded linear operator on (J, ‖ · ‖J ) and ‖δJ,A,B‖ ≤‖δA,B‖ for each norm ideal J in B(H).Let Jp denote the Schatten p-ideal, 1 ≤ p ≤ ∞; see for instance [9] or [13].The space Jp consists of compact operators K such that∑jspj(K) < ∞, where{sj(K)}j denotes the sequence of the singular values of K. For K ∈ Jp (1 ≤ p ≤∞), we set ‖K‖p =(∑jspj (K))1/p, where, by convention, ‖K‖∞ = s1(K) is theusual operator norm of K. Then (Jp, ‖ · ‖p) is a norm ideal. Moreover, (J2, ‖ · ‖2)is a Hilbert space with inner product defined by 〈X,Y 〉 = tr(Y ∗X) (X,Y ∈ J2),where tr denotes the usual trace functional.For simplicity of notation, we write δp,A,B (respectively Mp,A,B) insteadof δJp,A,B (respectively MJp,A,B) and δp,A instead of δJp,A,A.Stampfli [15] has given the elegant formula (1) for the norm of a gen-earlized derivation on B(H). Fialkow [7] has given an example of an operatorA ∈ B(H) such that ‖δ2,A‖ < ‖δA‖. This leads us to search for the relation be-tween the norms of ‖δA,B‖ and ‖δJ,A,B‖. It is true that for certain norm idealssuch as the compact operators or the trace class operators we do have equality,use duality or see [8].In order to state our results in detail, we first recall some notations andresults from the literature. Let E be a complex Banach space. For A ∈ B(E),let σ(A), σap(A) and r(A) denote respectively the spectrum, approximate pointspectrum and spectral radius of A. Recall that a complex number λ ∈ σap(A) ifthere exists a unit sequence {xn}n ⊆ E such that limn ‖(A − λ)xn‖ = 0. Sincethe boundary of σ(A) is contained in σap(A), then ‖A‖ ∈ σ(A) if and only if‖A‖ ∈ σap(A).The (algebraic) numerical range of A is defined byV (A) = {Φ(A) : Φ ∈ B(E)∗ and ‖Φ‖ = Φ(I) = 1},and the numerical radius of A is defined by v(A) = sup{|λ| : λ ∈ V (A,B(E))}.Note that V (A) is a compact convex subset of the plane and σ(A) ⊆ V (A)Generalized derivations and norm equality in normed ideals 109[4]. If E = H is a complex Hilbert space, then from [10], it turns out that thenorm ‖A‖ lies in W (A) = V (A) if and only if ‖A‖ lies in σap(A). An operatorT ∈ B(E) is said to be of class σ (respectively a normaloid operator) if r(T ) = ‖T‖(respectively v(T ) = ‖T‖). For any operator T ∈ B(H), r(T ) = ‖T‖ if and onlyif v(T ) = ‖T‖ see ([10]).The numerical range of a generalized derivation on norm ideals in B(E)was studied by several authors, see for instance [11] or [14]. In [14] S. Shawconsidered generalized derivations δJ,A,B acting on subspaces (J, ‖ · ‖J) of B(E)(E : Banach space) which satisfies axioms like those of norm ideals. He showedthe following equality:(2) V (δJ,A,B) = V (A)− V (B).Let K be a nonempty bounded subset of the plane. The diameter ofK is defined by diam(K) = supα,β∈K |α − β|. For A,B in B(H), we see fromabove that v(δJ,A,B) = sup{|α − β| : α ∈ V (A) and β ∈ V (B)}. On the otherhand, it turns out [6] that σ(δJ,A,B) = σ(A) − σ(B). Hence we deduce thatr(δJ,A,B) = sup{|α − β| : α ∈ σ(A) and β ∈ σ(B)}.In the following we denoted(A,B) = inf{‖A− λI‖+ ‖B − λI‖ : λ ∈ C}This is the norm of δA,B . Note that by a compactness argument there existsµ ∈ C such that d(A,B) = ‖A− µI‖+ ‖B − µI‖.Definition 0.1.1. An operator A ∈ B(H) is S-universal if ‖δJ,A‖ = d(A,A) = 2 inf{‖A−λI‖ :λ ∈ C} for each norm ideal J .2. A generalized derivation is said to be S-universal if ‖δJ,A,B‖ = d(A,B) foreach norm ideal J .The concept of a S-universal operator was introduced by L. Fialkow [7],who studied criteria for S-universality and posed several questions in this con-text. The S−universality of an operator was studied in [2]. The present paperstudies the S− universality of a generalized derivation. More precisely we shallbe concerned with equality ‖δA,B‖ = ‖δJ,A,B‖ for all normed ideals J .The main result of this paper is the following theorem:Theorem 0.2. Let A,B ∈ B(H) be non-zero. Let J be a norm ideal ofB(H). Then the following conditions are equivalent:1. ‖δ2,A,B‖ = d(A,B);110 M. Barraa2. There exists µ ∈ C such that M2,(A−µI)∗,(B−µI) is a normaloid operator.3. r(δJ,A,B) = d(A,B) (δJ,A,B is a σ operator);4. v(δJ,A,B) = d(A,B) (δJ,A,B is a normaloid operator);5. ‖δJ,A,B‖ = d(A,B) (δA,B is S-universal).If any one of this conditions is satisfied then r(A−µI) = ‖A−µI‖ and r(B−µI) =‖B − µI‖ where µ ∈ C such that d(A,B) = ‖A− µI‖+ ‖B − µI‖.1. Proof of the main result.P r o o f. 1) ⇒ 2) Assume that ‖δ2,A,B‖ = d(A,B) = ‖A − µ‖+ ‖B − µ‖.Since δ2,A,B = δ2,A−µ,B−µ = L2,A−µ −R2,B−µ, it follows that‖L2,A−µ −R2,B−µ‖ = ‖A− µ‖+ ‖B − µ‖.On the other hand, ‖L2,A−µ‖ = ‖A− µ‖ and ‖R2,B−µ‖ = ‖B − µ‖. Hence‖L2,A−µ −R2,B−µ‖ = ‖L2,A−µ‖+ ‖R2,B−µ‖.Without loss of generality we may assume that µ = 0, and then ‖L2,A −R2,B‖ = ‖L2,A‖+‖R2,B‖. By theorem 1 of [2], this is equivalent to ‖L2,A‖‖R2,B‖ ∈W (−L2,A∗R2,B). So‖L2,A‖‖R2,B‖ ≤ v(−L2,A∗R2,B) ≤ ‖L2,AR2,B‖ ≤ ‖L2,A‖‖R2,B‖.Thus −L2,A∗R2,B is a normaloid operator.2)⇒ 3) We know that J2 is a Hilbert space. In this case the condition 2)implies that ‖L2,A‖‖R2,B‖ ∈ σ(−L2,A∗R2,B). But σ(−L2,A∗R2,B) = −σ(A∗)σ(B)see [5] and ‖L2,A‖‖R2,B‖ = ‖A‖‖B‖. So there exist α ∈ σ(A) and β ∈ σ(B)such that ‖A‖‖B‖ = −αβ (α: complex conjugate of α). Since |α| ≤ ‖A‖ and|β| ≤ ‖B|, then one can find θ ∈ R such that α = ‖A‖eiθ and β = −‖B‖eiθ. Sor(δ2,A,B) = sup{|λ− µ| : λ ∈ σ(A), µ ∈ σ(B)} ≥ |α− β| = ‖A‖+ ‖B‖ = d(A,B).But σ(δ2,A,B) = σ(δJ,A,B) see [7]. So r(δJ,A,B) = d(A,B).3)⇒ 4) This is obvious.4) ⇒ 5) By the inequality v(δJ,A,B) ≤ ‖δJ,A,B‖ ≤ ‖δA,B‖ we see thatv(δJ,A,B) = d(A,B) imply that ‖δJ,A,B‖ = d(A,B).5)⇒ 1) Just take J = J2 The Hilbert Schmidt class ‖δ2,A,B‖ = ‖δA,B‖.Generalized derivations and norm equality in normed ideals 111From the proof of 1) imply 2), we see that if d(A,B) = ‖A−µI‖+‖B−µI‖then r(A− µI) = ‖A− µI‖ and r(B − µI) = ‖B − µI‖. 2. Examples and remarks.Remark 2.1. The ideal J2 in the condition 2) can’t be remplaced byB(H). The following example shows that MA,B can be normaloid and no othercondition in the theorem is satisfied.Example 2.2. Let A=(0 10 0)and B=(0 01 0). Then AB=(1 00 0).Easy computation gives W (AB) ⊂ V (MA,B) and so‖MJ,A,B ‖=‖ A ‖‖ B ‖= 1 = v(MA,B).But δJ,A,B is nilpotent for any ideal J, hence r(δJ,A,B) = 0. Note that v(A) =v(B) =12and v(δJ,A,B) = 1. It is also easy to show that ‖δA,B‖ = 2 and‖δ2,A,B‖ =√2.The following corollary summarizes some results from the third section of[2].Corollary 2.3. Let A ∈ B(H) non-zero. The Following conditions areequivalent:(1) ‖δ2,A‖ = ‖δA‖;(2) There exists µ ∈ C such that M2,(A−µI)∗,(A−µI) is a normaloid operator.(3) diam(σ(A)) = r(δJ,A) = ‖δA‖; (δJ,A is a σ operator.)(4) diamW (A) = v(δJ,A) = ‖δA‖; (δJ,A is a normaloid operator.)(5) ‖δJ,A‖ = ‖δA‖ ( A is a S-universal operator).Recently Timoney in [16] and [17], obtained a couple of general formulasfor the norm of an elementary operators. But it seems that this formulas throwno light on the norms of restrictions of such operators.REFERENCES[1] Y. A. Abramovich, C. D. Aliprantis, O. Burkinshaw. The Dau-gavet equation in uniformly convex Banach spaces. J. Func. Anal. 97 (1991),215–230.[2] M. Barraa, M. Boumazgour. Inner Derivations and Norm Equality. Proc.Amer. Math. Soc. 130 (2001), 471–476.112 M. Barraa[3] M. Boumazgour. An estimate for the norm of a derivation on a norm ideal.Linear and Multilear Algebra 54, 5 (2006), 321–327.[4] F. F. Bonsall, J. Duncan. Numerical ranges I,II. Cambridge Univ. Press,1973.[5] A. Brown, C. Pearcy. Spectra of tensor products of operators. Proc. Amer.Math. Soc. 17 (1966), 162–166.[6] A. Brown, C. Pearcy. On the spectra of derivations on norm ideals.Preprint.[7] L. Fialkow. A note on norm ideal and the operator X −→ AX − XB.Israel. J. Math. 32 (1979), 331–348.[8] L. Fialkow. Elementary operators & Applications. Proceeding of the inter-national workshop, 1991.[9] I. C. Goheberg, M. G. Krein. Introduction to the theory of linear non-selfadjoint operators. Trans. Math. Monographs vol. 18, Amer. Math. Soc.,Providence, RI, 1969.[10] P. R. Halmos. A Hilbert space Problem Book. Van Nostrand, Princeton,1970.[11] J. Kyle. Numerical ranges of derivations. Proc. Edinburgh. Math. Soc. 21(1978), 33–39.[12] C. L. Lin. The unilateral shift and norm equality for bounded linear oper-ators. Proc. Amer. Math. Soc. 127 (1999), 1693–1696.[13] R. Schatten. Norm ideals of completely continuous operators. Springer-Verlag, Berlin, 1960.[14] S. Y. Shaw. On numerical ranges of generalized derivations and relatedproperties. J. Austral. Math. Soc. Ser. A 36 (1984), 134–142.[15] J. Stampfli. The norm of a derivation, Pacific J. Math. 33 (1970), 737–747.[16] R. M. Timoney. Computing the norm of elementary operators. Illinois.J. Math. 47 (2003), 1207–1226.[17] R. M. Timoney. Some formulae for the norm of elementary operators.J. Operator Theory 57, 1 (2007), 121–145De´partement de Mathe´matiquesFaculte´ des SciencesSemlalia B.P: 2390 Marrakech, Maroce-mail: barraa@hotmail.com Received April 24, 2011

Generalized Derivations and Norm Equality in Normed Ideals

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Generalized Derivations and Norm Equality in Normed Ideals

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