2000 Mathematics Subject Classification: Primary 47B47, 47B10; Secondary 47A30.Let H be a separable infinite dimensional complex Hilbert space and let L(H) denote the algebra of all bounded linear operators on H into itself. Given A ∈ L(H), the derivation δA  :  L(H)→ L(H) is defined by δA(X) = AX-XA. In this paper we prove that if A is an n-multicyclic hyponormal operator and T is hyponormal such that AT = TA, then || δA(X)+T|| ≥ ||T|| for all X ∈ L(H). We establish the same inequality if A is a finite operator and commutes with normal operator T. Some related results are also given

Bouali, Said

Bouhafsi, Youssef

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica Math. J. 32 (2006), 31–38ON THE RANGE AND THE KERNEL OF DERIVATIONSSaid Bouali and Youssef BouhafsiCommunicated by L. TzafririAbstract. Let H be a separable infinite dimensional complex Hilbertspace and let L(H) denote the algebra of all bounded linear operators onH into itself. Given A ∈ L(H), the derivation δA : L(H) −→ L(H) isdefined by δA(X) = AX − XA. In this paper we prove that if A is an n-multicyclic hyponormal operator and T is hyponormal such that AT = TA,then ‖δA(X)+T‖ ≥ ‖T‖ for all X ∈ L(H). We establish the same inequalityif A is a finite operator and commutes with normal operator T . Some relatedresults are also given.1. Introduction. Let H be an infinite dimensional complex Hilbertspace and let L(H) denote the algebra of all bounded linear operators actingon H. If A ∈ L(H), then the inner derivation induced by A is the operator δAdefined on L(H) by δA(X) = AX − XA. By finite operator we shall mean abounded linear operator A on H such that(1) ‖δA(X) + I‖ ≥ 12000 Mathematics Subject Classification: Primary 47B47, 47B10; Secondary 47A30.Key words: Finite operator, n-multicyclic hyponormal operator.32 Said Bouali and Youssef Bouhafsifor every X ∈ L(H). As stated in [11] J. P. Williams proved that the classof finite operators contains every normaloid (i.e., the spectral radius r(A) of Aequals the norm of A), every operator with compact direct summand and theentire C∗-algebra generated by each of its members. The purpose of this paper isto investigate this class of operators to give natural generalizations of the norminequality (1). The basic tools in the main results is to use Anderson’s inequalityfor normal operators [1] and the Berberian extension Theorem [12, p. 15]. Thepresent paper is organized as follows. In Theorem 2.2 we initiate a new approachto extend this results to certain intertwining nonnormal operators A and T whereA is an n-multicyclic hyponormal operator and requiring that T is hyponormal.The point of view about finite operators is developed in Theorem 2.3, in which wegive a natural generalization of the inequality (1). Using a very simple argumentwe show in Theorem 2.4 that if A satisfies a quadratic polynomial, then A is afinite operator and that A∗ 6∈ R(δA)W, where R(δA)Wis the weak closure of therange R(δA) of δA.In addition to the notation already introduced, we shall use the followingfurther notation. Let K(H) be the ideal of all compact operators in L(H) andlet C(H) denote the Calkin algebra L(H)/K(H). For X ∈ L(H), let [X] denotethe projection of L(H) onto the Calkin algebra. We shall denote the kernel, theorthogonal complement of the kernel, the range of X by ker(X), ker(X)⊥ andR(X) respectively. The spectrum, the approximate point spectrum and the pointspectrum of X will be denoted by σ(X), σap(X) and σp(X), and the restrictionof X to an invariant subspace M will be denoted by X|M .Given A ∈ L(H), there exists a Hilbert space H◦ ⊃ H and an isometric ∗-isomorphism A −→ A◦ such that σ(A) = σ(A◦) and σap(A) = σap(A◦) = σp(A◦).This is the Berberian extension Theorem [12].2. Main results.Definition 2.1 [11]. Let A ∈ L(H). We say that A is a finite operatorif,‖AX −XA + I‖ ≥ 1,for all X ∈ L(H).Definition 2.2. Let A ∈ L(H). The operator A is said to be n-multicyclic if there are n vectors g1, g2, · · · , gn ∈ H, called generating vectors,such that {f(A)gi : f ∈ R(σ(A)), 1 ≤ i ≤ n} has span dense in H, whereR(σ(A)) denotes the rational functions analytic on σ(A).On the range and the kernel of derivations 33Theorem 2.1 [2]. If A is an n-multicyclic hyponormal operator, then[A∗, A] is in trace class, and tr([A∗, A]) ≤npiω(σ(A)), where ω is planar Lebesguemeasure.Theorem 2.2. Let A ∈ L(H). If A is an n-multicyclic hyponormaloperator, then for every hyponormal operator T such that AT = TA, we have‖AX −XA + T‖ ≥ ‖T‖for all X ∈ L(H).P r o o f. Let T be a hyponormal operator in L(H) such that AT = TA.We have r(T ) = ‖T‖, then it is enough to show that‖AX −XA + T‖ ≥ |λ|for all X ∈ L(H) and all λ ∈ σ(T ). It is well known that T enjoys the propertythat σ(T ) = σp(T ) ∪ σe(T ) (see [5]).Let λ ∈ σ(T ), we will consider two cases for the location of λ.Case 1. Assume that λ ∈ σp(T ). We shall divide this case into twodifferent steps.(i) If λ ∈ σp(T ) such that dimker(T−λ) < ∞. Since A commutes with T ,it follows that the subspace Eλ = ker(T − λ) is invariant by T and A. Moreover,A/Eλ is normal hence Eλ reduces A and T simultaneously [9, p. 514]. Then withrespect to the orthogonal decomposition H = Eλ ⊕ E⊥λ , the operators A and Tcan be respectively represented asA =(B 00 C)and T =(λ 00 ∗).Let X ∈ L(H) have the matrix representation X =(Y ZR S), we get‖AX −XA + T‖ =∥∥∥(BY − Y B + λ ∗∗ ∗)∥∥∥.Since the norm of an operator matrix always dominates the norm of its diagonalpart [7, p. 82], it follows that‖AX −XA + T‖ ≥ ‖BY − Y B + λ‖34 Said Bouali and Youssef BouhafsiA is a finite operator because A is hyponormal [11], hence B thus. Then weobtain‖BY − Y B + λ‖ ≥ |λ|Consequently we have‖AX −XA + T‖ ≥ |λ|for all X ∈ L(H), and all λ ∈ σp(T ) such that dimker(T − λ) < ∞.(ii) If λ ∈ σp(T ) such that dimker(T −λ) = ∞. Since T is a hyponormaloperator, then dimker(T − λ)∗ = ∞. It follows that T − λ is not a Fredholmoperator, hence λ ∈ σe(T ) (see the case 2.).Case 2. Assume that λ ∈ σe(T ). Also, we will divide this case into twosteps.(i) T has no isolated eigenvalues of finite multiplicity.The condition AT = TA implies that [A][T ] = [T ][A]. Since A is an n-multicyclichyponormal operator, it follows from the Theorem 2.1 that [A] is normal. An-derson’s result [1] applied to the Calkin algebra insures that‖AX −XA + T‖ ≥ ‖[A][X] − [X][A] + [T ]‖ ≥ ‖[T ]‖for all X ∈ L(H). On the other hand, since T is hyponormal and has no isolatedeigenvalues of finite multiplicity, one obtains from Remark [5, p. 186] that ‖[T ]‖ =r([T ]). Hence‖[A][X] − [X][A] + [T ]‖ ≥ |λ|for all X ∈ L(H). It follows that‖AX −XA + T‖ ≥ |λ|for all X ∈ L(H), as desired.(ii) If T has isolated eigenvalues of finite multiplicity. We consider thesubspace E =⊕µ∈β(T ) ker(T−µ), where β(T ) is the set of all isolated eigenvaluesof T with finite multiplicity. Since T/E is a normal operator then E reduces T .With respect to the decomposition H = E ⊕E⊥, we haveT =(T1 00 T2).It follows from an application of Anderson’s result [1] and the Theorem 2.1 that‖AX −XA + T‖ ≥ ‖[A][X] − [X][A] + [T ]‖ ≥ ‖[T ]‖.On the range and the kernel of derivations 35Moreover, λ ∈ σe(T ) if and only if λ ∈ σe(T2). By hypothesis we have λ ∈σe(T ) = σe(T2). It is easy to check thatinf{∥∥∥∥(K1 + T1 K2K3 T2 + K4)∥∥∥∥ , K1,K2,K3,K4 compacts}≥≥ inf{‖T2 + K4‖, K4 compact}Then it follows immediately‖[T ]‖ ≥ ‖[T2]‖.Since T2 has no isolated eigenvalues of finite multiplicity, then by using the Re-mark [5, p. 186] we have ‖AX −XA + T‖ ≥ |λ|. Whence‖AX −XA + T‖ ≥ |λ|For all X ∈ L(H) and all λ ∈ σe(T ). Finally, we conclude that‖AX −XA + T‖ ≥ |λ|For all X ∈ L(H) and all λ ∈ σ(T ), and the proof is complete. As a special case we get the following Corollary.Corollary 2.1. Let A, T ∈ L(H), such that A quasi-normal operator, Thyponormal and AT = TA. Then‖AX −XA + T‖ ≥ ‖T‖for all X ∈ L(H).P r o o f. Since A is a quasi-normal operator, it follows from [6] that A =N + K, where N is a normal and K is a compact. Hence, by using the sameargument as in the above theorem, we get the desired inequality. Theorem 2.3. Let A and T be commuting operators such that A is afinite operator and T is normal. Then‖AX −XA + T‖ ≥ ‖T‖for all X ∈ L(H).P r o o f. Let λ ∈ σp(T ) and let E be the subspace E = ker(T − λ). SinceA commutes with T it follows from Fuglede-Putnam’s theorem [8] that E reduces36 Said Bouali and Youssef BouhafsiA and T simultaneously. Hence, with respect to the decomposition H = E⊕E⊥,we haveA =(B 00 ∗)and T =(λ 00 ∗).Let X ∈ L(H) have the representation X =(Y ∗∗ ∗)∈ L(H). Then,‖AX −XA + T‖ =∥∥∥∥(BY − Y B + λ ∗∗ ∗)∥∥∥∥ ≥ ‖BY − Y B + λ‖Since B is a finite operator, it follows that‖AX −XA + T‖ ≥ |λ|,for all X ∈ L(H) and all λ ∈ σp(T ).Using the Berberian extension Theorem, we have that A◦ is finite , T ◦ isnormal and A◦T ◦ = T ◦A◦. Since, σp(T◦) = σap(T◦) = σap(T ) = σ(T ), it followsfrom the first part that‖AX −XA + T‖ ≥ |λ|for all X ∈ L(H) and all λ ∈ σ(T ). Hence,‖AX −XA + T‖ ≥ ‖T‖for all X ∈ L(H). This completes the proof. Theorem 2.4. Let A ∈ L(H). If A satisfies some quadratic polynomial,then A is a finite operator and A∗ 6∈ R(δA)W.P r o o f. Suppose that A satisfies A2 − 2αA + β = 0, hence (A− α)2 is anormal operator. Then, by Putnam’s result [10] we may write A− α = N ⊕M ,where N is normal and M =(B C0 −C), with B normal and C is an injectivepositive operator such that BC = CB. Therefore, A = (N + αI) ⊕ (M + αI).Then for linear operator X =(Y ZR S)we haveAX −XA + I =(NY − Y N + I ∗∗ ∗).On the range and the kernel of derivations 37Since the norm of operator matrix always dominates the norm of its diagonalpart [7, p. 82] one obtains‖AX −XA + I‖ ≥ ‖NY − Y N + I‖.Hence it follows from Williams’s result [11] on normal operators that‖AX −XA + I‖ ≥ 1for all X ∈ L(H). Let us assume that A∗ ∈ R(δA)W. An easy calculation leadsus to A∗A ∈ R(δA)W. Since R(δA)Wcontains nonzero positive operator [3], itfollows that A = 0. Remark 2.1. The above theorem is due to J. P. Williams [11], howeverwe proved it by other method that this used.Another interesting class of operators for which the Theorem 2.3 is sat-isfied is the class of operators A such that A∗A and A + A∗ commute. It is wellknown that this class has the property that r(A) = ‖A‖ (see [4]).REFERE NCES[1] J. H. Anderson. On normal derivations. Proc. Amer. Math. Soc. 38(1973), 135–140.[2] C. A. Berger, B. I. Shaw. Selfcommutators of multicyclic hyponormaloperators are always trace class. Bull. Amer. Math. Soc. 79 (1973), 1193–1199.[3] S. Bouali, J. Charles. Extension de la notion d’ope´rateur d-syme´triqueII. Linear Algebra Appl. 225 (1995), 175–185.[4] S. L. Campbell, R. Gellar. Linear operators for which T ∗T and T +T ∗commute. Proc. Amer. Math. Soc. 60 (1976), 197–202.[5] P. A. Fillmore, J. G. Stampfli, J. P. Williams. On the essentialnumerical range, the essential spectrum, and a problem of Halmos. ActaSci. Math. 33 (1972),179–192.[6] M. I. Gil. Inner bound for the spectrum of quasi-normal operators. Proc.Amer. Math. Soc. 131 (2003), 3737–3746.38 Said Bouali and Youssef Bouhafsi[7] I. C. Gohberg, M. G. Krein. Introduction to the theory of linear non-self adjoint operators. Transl. Math. Monographs, vol 18, A. Math. Soc.,Providence, R.I., 1969.[8] P. R. Halmos. A Hilbert space problem book. Springer-Verlag, New York,1982.[9] R. L. Moore, D. D. Rogers. Note on intertwining M -hyponormal op-erators. Proc. Amer. Math. Soc. 83 (1981), 514–516.[10] C. H. Putnam. Range of normal and subnormal operators. MichiganMath. J. 18 (1971), 33–36.[11] J. P. Williams. Finite operators. Proc. Amer. Math. Soc. 26 (1970), 129–136.[12] D. Xia. Spectral theory of hyponormal operators. Birkhauser Verlag(Basel), 1983.Department of MathematicsFaculty of SciencesB.P.133, Ke´nitra , Moroccoe-mail: bouali.said1@caramail.come-mail: ybouhafsi@yahoo.frReceived July 07, 2005Revised September 4, 2005

On the Range and the Kernel of Derivations

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On the Range and the Kernel of Derivations

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