2000 Mathematics Subject Classification: Primary: 47B47, 47B10; secondary 47A30.Let H be an infinite-dimensional complex Hilbert space and let A, B ∈ L(H), where L(H) is the algebra of operators on H into itself. Let δAB: L(H) → L(H) denote the generalized derivation δAB(X) = AX − XB. This note will initiate a study on the class of pairs (A,B) such that [‾(R(δAB))] = [‾(R(δB*A*))]; i.e. [‾(R(δAB))] is self-adjoint

Bouali, S.

Ech-chad, M.

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Serdica Math. J. 34 (2008), 557–562GENERALIZED D-SYMMETRIC OPERATORS IS. Bouali, M. Ech-chadCommunicated by L. TzafririAbstract. Let H be an infinite-dimensional complex Hilbert space and letA, B ∈ L(H), where L(H) is the algebra of operators on H into itself. LetδAB : L(H) → L(H) denote the generalized derivation δAB(X) = AX−XB.This note will initiate a study on the class of pairs (A, B) such thatR(δAB) =R(δB∗A∗); i.e. R(δAB) is self-adjoint.Introduction. Let L(H) the algebra of all bounded operators on aninfinite dimensional complex Hilbert space H. The generalized derivation oper-ator δAB associated with (A,B), defined on L(H) by δAB(X) = AX −XB wassystematically studied for the first time in [6]. The properties of such operatorshave been studied extensively (see for example [2, 5, 8, 9, 10]).The D-symmetric operators (A is D-symmetric if R(δA) is self-adjoint,where R(δA) is the closure of the range R(δA) of δA in the norm topology ) werestudied by J. H. Anderson, J. W. Bunce, J. A. Deddens and J. P. Williams [1],S. Bouali and J. Charles [3, 4] and J. G. Stampfli [8].2000 Mathematics Subject Classification: Primary: 47B47, 47B10; secondary 47A30.Key words: Generalized derivation, self-adjoint derivation ranges, D-symmetric operators.558 S. Bouali, M. Ech-chadWe consider the class of pairs (A,B) such that R(δAB) is self-adjoint,we call such pairs D-symmetric. In this work we extend the results of the D-symmetric operators to D-symmetric pairs.In the first part we give some properties and characterizations whichconcern the D-symmetric pairs. The second part contains a description of thesets:C(A,B) = {C ∈ L(H), CL(H) + L(H)C ⊂ R(δAB)}andI(A,B) = {Z ∈ L(H), ZR(δAB) +R(δAB)Z ⊂ R(δAB)}which generalize those introduced by J. P. Williams in [10].Notations.1. Let K(H) be the ideal of all compact operators. For A ∈ L(H), let [A]denote the coset of A in the Calkin algebra C(H) = L(H)/K(H).2. C1(H) is the ideal of trace class operators.3. For A,B ∈ L(H), R(δAB)Udenotes the ultraweak closure of R(δAB),and L(H)′U denotes the bounded linear forms in ultraweak topology.4. Let M be a subspace of L(H). We denote the orthogonal of M in theduality L(H), L(H)′ by M o.5. For g and ω two vectors in H, we define g ⊗ ω ∈ L(H) as follows:g ⊗ ω(x) =< x,ω > g for all x ∈ H.1. Properties of D-symmetric Pairs.Definition 1.1. Let A,B ∈ L(H).(1) If R(δAB) is self-adjoint i.e. R(δAB) = R(δB∗A∗), we say that (A,B)is D-symmetric pair of operators. We denote the set of such pairs by GD(H).(2) Let δ[A][B] the generalized derivation operator defined on C(H) byδ[A][B]([X]) = [δAB(X)]. If R(δ[A][B]) is self-adjoint i.e. R(δ[A][B]) = R(δ[B∗][A∗]),we say that ([A], [B]) is D-symmetric in C(H).Lemma 1.1. If A,B ∈ L(H), thenR(δAB)0 ' R(δAB)0 ∩ K(H)0 ⊕ ker (δBA) ∩ C1(H).Generalized D-symmetric Operators I 559The proof of Lemma 1.1 is the same as the proof of Theorem 3 in [11].Theorem 1.1. For A, B ∈ L(H) the following are equivalent:(1). (A,B) is D-symmetric;(2). a. ([A], [B]) is D-symmetric in C(H), andb. BT = TA implies BT ∗ = T ∗A for all T ∈ C1(H);(3). c. ([A], [B]) is D-symmetric in C(H), andd. R(δAB)U= R(δB∗A∗)U.P r o o f. Note that R(δAB)Uis self-adjoint if and only ifR(δAB)0∩L(H)′Uis self-adjoint. Using Lemma 1.1 we haveR(δAB)0 ∩ L(H)′U ' ker (δBA) ∩ C1(H).Consequently we obtain: R(δAB)Uis self-adjoint if and only if ker (δBA)∩C1(H)is self-adjoint. Thus (2) ⇔ (3).The equivalence of (1) and (2) is a consequence of Lemma 1.1. Theorem 1.2. Let A,B ∈ L(H). If there exists λ ∈ IC such that (B −λ)(A − λ) = (A − λ)2 = 0, A − λ 6= 0 and B − λ 6= 0, then (A,B) is notD-symmetric.P r o o f. Since for all λ ∈ IC, R(δAB) = R(δ(A−λ)(B−λ)), we may assumewithout loss of generality that λ = 0. The condition A∗A 6= 0 (A 6= 0) impliesthat there exists an vector f = Ah 6= 0, such that A∗f 6= 0. Then Bf = 0. SinceA∗B∗ = 0, we choose g 6= 0 such that A∗g = 0. We put A∗f = ω;〈ω, f〉 = 〈A∗f, f〉 = 〈f,Af〉 = 〈f,A2h〉 = 0i.e. ω and f are orthogonal. If X = ‖ω‖−2(g⊗ ω) and Y ∈ L(H), then it followsthat:〈(B∗X −XA∗)f, g〉 = 〈B∗Xf, g〉 − 〈XA∗f, g〉= 〈0, g〉 − 〈Xω, g〉= −〈g, g〉= −‖g‖2and〈(AY − Y B)f, g〉 = 〈Y f,A∗g > − < 0, g〉 = 0.Suppose that B∗X − XA∗ ∈ R(δAB)U. Then there exists a net (Yα)α ⊂ L(H)such that, for all x and y in H, we have:〈(AYα − YαB)x, y〉 −→ 〈(B∗X −XA∗)x, y〉.560 S. Bouali, M. Ech-chadSo that,0 = 〈(AYα − YαB)f, g〉 −→ 〈(B∗X −XA∗)f, g〉 = −‖g‖2.It follows that g = 0; this proves that B∗X −XA∗ /∈ R(δAB)U. Consequently weobtain that (A,B) is not D-symmetric by Theorem 1.1. Theorem 1.3. If H is separable, then GD(H) is not norm-closed in(L(H))2.P r o o f. Let {en}n≥1 be an orthonormal basis for H. Define a sequenceof operators (Sn)n≥1 as follows:Sn(ek) =1ne2, if k = 1;ek+1, if k ≥ 2.Corollary 3 in [7] asserts that for every n ≥ 1 K(H) ⊂ R(δSn). It follows from[11, Corollary 1, p. 277] that {Sn}′ ∩ C1(H) = {0}, then Theorem 1.1 impliesthat (Sn, Sn) ∈ GD(H) for all n ≥ 1. LetS(ek) ={0, if k = 1;ek+1, if k ≥ 2.It is clear that ‖(Sn, Sn) − (S, S)‖ −→ 0. Let f = e1 + e2, ω = e3 and g = e1.Since S∗f = 0, Sf = ω and Sg = 0, It follows from the proof of Theorem 1.2that (S∗, S∗) is not D-symmetric. Thus (S, S) /∈ GD(H), which completes theproof. 2. Properties and Descriptions of C(A, B) and I(A, B). Con-sider the natural closed subalgebras of L(H) associated with (A,B):C(A,B) = {C ∈ L(H), CL(H) + L(H)C ⊂ R(δAB)}andI(A,B) = {Z ∈ L(H), ZR(δAB) +R(δAB)Z ⊂ R(δAB)}It is clear that; if R(δAB) is norm-dense in L(H), I(A,B) = C(A,B) = L(H) (forexample A = 2B = 2I). Thus C(A,B) 6= {0} and I(A,B) contains non-scalaroperators in general.Theoren 2.1. If (A,B) is D-symmetric, then:Generalized D-symmetric Operators I 561ı. C(A,B) and I(A,B) are norm closed C∗−algebras in L(H);ıı. C(A,B) is a two-sided ideal of I(A,B).P r o o f. ı. It is clear that C(A,B) and I(A,B) are norm closed algebrasin L(H). Since R(δAB) is self-adjoint, C(A,B) and I(A,B) are C∗−algebras.ıı. If Z ∈ I(A,B) and C ∈ C(A,B), then for all X ∈ L(H) we have:X(CZ) = (XC)Z ∈ R(δAB)Z ⊂ R(δAB),and (CZ)X = C(ZX) ∈ R(δAB). Thus C(A,B) is a right ideal of I(A,B). SinceC(A,B) and I(A,B) are C∗−algebras, C(A,B) is a two-sided ideal of I(A,B). Lemma 2.1. Let A, B ∈ L(H), then;I(A,B) = {Z ∈ L(H), δZ(A)L(H) + L(H)δZ(B) ⊂ R(δAB)}.P r o o f. If Z ∈ I(A,B) and X ∈ L(H), thenδZ(A)X = ZδAB(X) − δAB(ZX), and XδZ(B) = δAB(X)Z − δAB(XZ).This implies that δZ(A)X ∈ R(δAB) and XδZ(B) ∈ R(δAB). ThusδZ(A)L(H) + L(H)δZ(B) ⊂ R(δAB).The reverse inclusion follows from the identities:ZδAB(X) = δZ(A)X + δAB(ZX), and δAB(X)Z = XδZ(B) + δAB(XZ).2Theorem 2.2. Let A, B ∈ L(H). If R(δAB) does not contain anynonzero positive operator, then C(A,B) = {0} and I(A,B) = {A}′ ∩ {B}′.P r o o f. If C ∈ C(A,B) then CC∗ ∈ R(δAB); consequently we haveC = 0. Thus C(A,B) = {0}.Let Z ∈ I(A,B), δZ(A)L(H) ⊂ R(δAB) and L(H)δZ(B) ⊂ R(δAB) byLemma 2.1.Consequently we obtain δZ(A)(δZ(A))∗ = (δZ(B))∗δZ(B) = 0. ThusZ ∈ {A}′ ∩ {B}′.Conversely; if Z ∈ {A}′ ∩ {B}′, then δZ(A) = δZ(B) = 0. It follows fromLemma 2.1 that Z ∈ I(A,B). 562 S. Bouali, M. Ech-chadREFERE NCES[1] J. H. Anderson, J. W. Bunce, J. A. Deddens, J. P. Williams. C*-algebras and derivation ranges. Acta Sci. Math. (Szeged), 40 (1978), 211–227.[2] J. H. Anderson, C. Foias. Properties which normal operators share withnormal derivation and related operators. Pacific J. Math., 61 (1976), 313–325.[3] S. Bouali, J. Charles. Extension de la notion d’ope´rateurs d-syme´triquesI. Acta Sci. Math. (Szeged) 58 (1993), 517–525.[4] S. Bouali, J. Charles , Extension de la notion d’ope´rateurs d-syme´triquesII. Linear Algebra And Its Applications 225 (1995), 175–185.[5] D. A. Herrero. Approximation of Hilbert space operators, I. Pitman,Advanced publishing program, Boston – Melbourne, 1982.[6] M. A. Rosenblum. On the operator equation BX−XA = Q. Duke Math.J. 23 (1956), 263–269.[7] C. Rosentrater. Compact operators and derivations induced by weightedshifts. Pacific J. Math. 104 (1983), 465–470.[8] J. G. Stampfli. On self-adjoint derivation ranges. Pacific J. Math. 82(1979), 257–277.[9] J. G. Stampfli. The norm of a derivation. Pacific J. Math. 33 (1970),737–747.[10] J. P. Williams. Derivation ranges: Open problems. Topics in Modern Op-erator Theory, Birkhauser-Verlag, 1981, 319–328.[11] J. P. Williams. On the range of a derivation. Pacific J. Math. 38 (1971),273–279.Department of Mathematics and InformaticsFaculty of SciencesKe´nitra, B.P. 133 Ke´nitraMorocco.e-mail: said.bouali@yahoo.frm.echchad@yahoo.frReceived April 20, 2007Revised July 9, 2007

Generalized D-Symmetric Operators I

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Generalized D-Symmetric Operators I

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