Mathematica Bohemica

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1 Mathematica Bohemica Said Bouali; Youssef Bouhafsi On the range-kernel orthogonality of elementary operators Mathematica Bohemica, Vol. 140 (2015), No. 3, Persistent URL: Terms of use: Institute of Mathematics AS CR, 2015 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 140(2015) MATHEMATICA BOHEMICA No. 3, ON THE RANGE-KERNEL ORTHOGONALITY OF ELEMENTARY OPERATORS Said Bouali, Kénitra, Youssef Bouhafsi, Rabat (Received January 16, 2013) Abstract. Let L(H) denote the algebra of operators on a complex infinite dimensional Hilbertspace H. For A,B L(H),thegeneralizedderivation δ A,B andtheelementary operator A,B aredefinedby δ A,B (X)=AX XBand A,B (X)=AXB Xforall X L(H). Inthispaper,weexhibitpairs(A,B)ofoperatorssuchthattherange-kernel orthogonalityof δ A,B holdsfortheusualoperatornorm.wegeneralizesomerecentresults. Wealsoestablishsometheoremsontheorthogonalityoftherangeandthekernelof A,B with respect to the wider class of unitarily invariant norms on L(H). Keywords: derivation; elementary operator; orthogonality; unitarily invariant norm; cyclic subnormal operator; Fuglede-Putnam property MSC 2010: 47A30, 47A63, 47B15, 47B20, 47B47, 47B10 1. Introduction Let HbeacomplexinfinitedimensionalHilbertspace,andlet L(H)denotethe algebraofallboundedlinearoperatorsactingon Hintoitself.Given A,B L(H), wedefinethegeneralizedderivation δ A,B : L(H) L(H)by δ A,B (X) = AX XB, andtheelementaryoperator A,B : L(H) L(H)by A,B (X) = AXB X.Let δ A,A = δ A and A,A = A. In[1],Andersonshowsthatif Aisnormalandcommuteswith T,thenforall X L(H) (1.1) δ A (X)+T T, where istheusualoperatornorm. Inviewof[1],Definition1.2,theinequality (1.1)saysthattherange R(δ A )of δ A isorthogonaltoitskernel ker(δ A ),whichis justthecommutant {A} of A. 261

3 If Aand Barenormaloperatorssuchthat AT ) = TBforsome ) T L(H),notice thatifweconsidertheoperators A B, and on H H,thenforall X L(H)wehave ( 0 X 0 0 δ A,B (X)+T T. ( 0 T 0 0 Inequality(1.1)hasa A analogue. Thus,Duggal[6]provedthatif Aisanormal operatorsuchthat A (T) = 0forsome T L(H),thenforall X L(H)wehave A (X)+T T. The orthogonality of the range and the kernel of elementary operators with respect tothewiderclassofunitarilyinvariantnormson L(H)hasbeenconsideredbymany authors[3],[5],[6],[8],[10]and[11]. The purpose of this paper is to study the range-kernel orthogonality of the operators δ A,B and A,B. Wegivepairs (A,B)ofoperatorssuchthattherangeand thekernelof δ A,B areorthogonal. Weexhibitpairs (A,B)ofoperatorssuchthat R(δ A,B )isorthogonalto ker(δ A,B ). Weinvestigatetheorthogonalityoftherangeandthekernelof A,B innorm ideals. Related results on orthogonality for certain elementary operators are also given. Given X L(H),weshalldenotethekernel,theorthogonalcomplementofthe kernelandtheclosureoftherangeof Xby ker(x), ker (X),and R(X),respectively. Thespectrumof Xwillbedenotedby σ(x),and X Mwilldenotetherestriction of Xtoaninvariantsubspace M. 2. Main results Definition2.1. Let Ebeanormedlinearspaceand Cthecomplexnumbers. 1)Wesaythat x Eisorthogonalto y Eif x λy λy forall λ C. 2)Let Fand Gbetwosubspacesin E.If x+y y forall x Fandforall y G,then Fissaidtobeorthogonalto G. Remark2.1. Notethatif xisorthogonalto y,then yneednotbeorthogonalto x. This definition generalizes the idea of orthogonality in Hilbert space. Itisshownin[1]thatif F isorthogonalto G,and F, Gareclosedsubspaces of E,thenthealgebraicdirectsum F Gisaclosedsubspacein E. 262

4 Theorem2.1.Let A,B L(H).If Bisinvertibleand A B 1 1,then δ A,B (X)+T T forall X L(H)andforall T ker(δ A,B ). Proof. Let T L(H),suchthat AT = TB. Thisimpliesthat ATB 1 = T. Since A B 1 1,itfollowsfrom[11],Corollary1.4,that AYB 1 Y +T T forall Y L(H).Ifweset X = YB 1,thenweget AX XB +T T. Hence δ A,B (X)+T T forall T ker(δ A,B )andforall X L(H). Theorem2.2.Let A,B L(H).Ifeither 1) Aisanisometryandtheoperator Bisacontractionor 2) Aisacontractionand Bisco-isometric,then δ A,B (X)+T T forall X L(H)andforall T ker(δ A,B ). Proof. 1)Given T ker(δ A,B ),wehave Moreover, we see that δ A,B (T) = 0 T = A TB A T = A (A T)B. δ A,B (X)+T A (δ A,B (X)+T) = A,B(X) A T. Since Aisanisometryand Bisacontraction,itfollowsfrom[11],Corollary1.4, that δ A,B (X)+T A,B(X) A T A T A TB = T. Then, δ A,B (X)+T T forall X L(H). 2)Let T ker(δ A,B )and X L(H).Bytakingadjoints,observethat δ A,B (X)+T = δ B,A (X ) T. Since B isisometricand A isacontraction,theresultfollowsfromthefirstpart of the proof. AsanapplicationofTheorem2.2wehaveawellknownresult. 263

5 Corollary2.1.LetU, Vbeisometriessuchthat δ U,V (T) = 0forsome T L(H). Then δ U,V (X)+T T forall X L(H). Remark2.2.Let A,B L(H).If Aisanisometryand Bisacontraction,then R(δ A,B ) ker(δ A,B ) = {0}. Definition2.2([7]). Apropertwo-sidedideal J in L(H)issaidtobeanorm ideal if there is a norm on J possessing the following properties: i) (J, J )isabanachspace. ii) AXB J A X J B forall A,B L(H)andforall X J. iii) X J = X for Xarankoneoperator. Remark2.3.If (J, J )isanormideal,thenthenorm J isunitarilyinvariant,inthesensethat UAV J = A J forall A J andforallunitaryoperators U,V L(H). Corollary 2.2. Let (J, J )beanormidealand A,B L(H). If Aisan isometry and the operator B is a contraction, then δ A,B (X)+T J T J forall X J andforall T ker(δ A,B ) J. Theorem2.3.Let (J, J )beanormidealand A L(H).Supposethat f(a) is a cyclic subnormal operator, where f is a nonconstant analytic function on an open set containing σ(a). Then forall X J andforall T {A} J. δ A (X)+T J T J Proof. Let T J besuchthat AT = TA,thenwehave f(a)t = Tf(A)and Af(A) = f(a)a.since f(a)isacyclicsubnormaloperator,itfollowsfromyoshino s result[12] that T and A are subnormal. Therefore, every compact hyponormal operator is normal[2], hence T is normal. Consequently,AT = TAimpliesthatAT = T A.HenceweobtainthatR(T)and ker (T)reduces A,and A 0 = A/R(T)and B 0 = A/ker (T)arenormaloperators. 264

6 Let A = A 0 A 1 withrespectto H 0 = H = R(T) R(T), andlet A = B 0 B 1 withrespectto H 1 = H = ker (T) ker(t). Definethequasi-affinity T 0 : ker (T) R(T)bysetting T 0 x = Txforevery x ker (T). Thenitresults that δ A0,B 0 (T 0 ) = δ A 0,B 0 (T 0) = 0. Also,wecanwrite Tand Xon H 1 into H 0 as Consequently, we have T = ( ) T0 0, X = 0 0 ( X0 X 1 X 2 X 3 ( δ A (X)+T J = δa0,b 0 (X 0 )+T 0 J ) ). δ A0,B 0 (X)+T J. Since A 0 and B 0 arenormaloperators,weobtainfrom[4],theorem4,that δ A (X)+T J δ A0,B 0 (X 0 )+T 0 J T 0 J = T J. Remark 2.4. Let A L(H)andlet fbeananalyticfunctiononanopenset containing σ(a).if f(a)iscyclicsubnormaland Tisacompactoperatorsuchthat AT = TA,thenforall X L(H), δ A (X)+T T. Definition 2.3. Let A,B L(H)andlet J beatwo-sidedidealof L(H). Wesaythatthepair (A,B)possessestheFuglede-Putnamproperty PF(,J),if ker( A,B J) ker( A,B J). Theorem 2.4. Let A,B L(H). Ifthepair (A,B)possessesthe PF(,J) property, then A,B (X)+T J T J forall X J,andforall T ker( A,B ) J. Proof. Given T J suchthat ATB = T. Sincethepair (A,B)possesses the PF(,J)property, R(T)reduces A,and ker (T)reduces B,and A 0 = A R(T), B 0 = B ker (T)arenormaloperators. Let T 0 : ker (T) R(T)bethequasi-affinitydefinedbysetting T 0 x = Txfor each x ker (T). Thenwehave A0,B 0 (T 0 ) = 0 = A 0,B 0 (T 0). Let A = A 0 A 1 265

7 withrespectto H 0 = H = R(T) R(T), and B = B 0 B 1 withrespectto H 1 = H = ker (T) ker(t).let Xon H 1 into H 0 havethematrixrepresentation ( X0 X 1 X = X 2 X 3 ). Hence ( A,B (X)+T J = A0,B 0 (X 0 )+T 0 J. ) Itfollowsfrom[7]thatthediagonalpartofablockmatrixalwayshassmallernorm than that of the whole matrix. Consequently, we have ( A,B (X)+T J = A0,B 0 (X 0 )+T 0 J ) Since A 0 and B 0 arenormal,itresultsfrom[6],theorem2,that A0,B 0 (X 0 )+T 0 J. A,B (X)+T J A0,B 0 (X 0 )+T 0 J T 0 J = T J. The following corollaries are consequences of the above theorem. Corollary2.3.LetA,B L(H).Letsomeofthefollowingconditionsbefulfilled: 1) A,B L(H)suchthat Ax x Bx forall x H. 2) Aisinvertibleand Bsuchthat A 1 B 1. 3) Aisdominantand B ism-hyponormal. Thenwehave A,B (X)+T J T J forall X J andforall T ker( A,B ) J. Proof. Itissufficienttoshowthatthepair (A,B)hastheFuglede-Putnam property PF(,J)ineachoftheprecedingcases(inparticular(3)). 1)Itfollowsfrom[9],Lemma1,thatforall T ker( A,B ) J,wehave R(T) reduces Aand ker (T)reduces B,and A R(T),B ker (T)areunitaryoperators. Hence,itresultsthatthethepair (A,B)hastheproperty PF(,J). 2)Inthiscase,let A 1 = B 1 Aand B 1 = B 1 B,then A 1 x x B 1 x forall x H.Hence,theresultholdsdueto(1.1). 266

8 Corollary2.4. Let A,B L(H)besuchthatthepairs (A,A)and (B,B)have the PF(,J)property.If 1 σ(a)σ(b),then A,B (X)+T J T J forall X J,andforall T ker( A,B ) J. Proof. Itiswellknownthatif 1 σ(a)σ(b),thentheoperators A,B and B,A areinvertible.thus,asimplecalculationshowsthatthepair (A B,A B) possesses the PF(, J) property. Remark2.5.If Se n = ω n e n+1 isaunilateral(bilateral)weightedshift,then,it followsfrom[3]thatthepair (S,S)hastheproperty PF(δ,J)ifandonlyif ω k ω k+1...ω k+n 1 =. k Remark2.6.1)Let A,B L(H),then R( A,B ) ker( A,B ) = {0}ineachof the following cases: i) Aand Barenormal. ii) Aand Barecontraction. iii) A = Biscyclicsubnormal. iv) Aand B arehyponormal. 2)If A and Barehyponormal,then R( A,B ) ker( A,B ) = {0}. Corollary 2.5. Let A,B L(H). Then every operator in R( A B ) {ker( A B ) ker( A B )}isnilpotentofordernotgreaterthan 2,ineachof the following cases: 1) Anormaland Bisometric. 2) Anormaland Bcyclicsubnormal. 3) A cyclic subnormal and B co-isometric. Proof. On H H,let T betheoperatordefinedas T = calculationshowsthat T R( A B ) ker( A B )implies P R( A ) ker( A ); S R( B ) ker( B ); R R( B,A ) ker( B,A ); Q R( A,B ) ker( A,B ). ( ) P Q R S. Aroutine Hence, if A isnormaland B isisometric, it followsfrom[6], Corollary1, [11], ), Corollary1.4,that P = 0, S = 0and R = 0. Consequently,weobtain T = which ensures that T is nilpotent of order not greater than 2. By using a similar argument we get the desired result. ( 0 Q

9 Remark )NotethatCorollary2.5stillholdsifweconsidertheinner derivation δ A insteadof A. 2)Let π: L(H) L(H) K(H)denotetheCalkinmap.Set S = {T L(H): π(t) = T }. If A L(H)satisfiesoneofthefollowingconditions: i) A A AA iscompact; ii) A A Ior AA Iiscompact; then R(d A )isorthogonalto ker(d A ) S,where d A = δ A or d A = A. 3. Acommentandsomeopenquestions 1)Itisshownin[3]thatif Aisacyclicsubnormaloperator,then R(δ A )isorthogonalto {A},andthisorthogonalityfailsintheabsenceofthehypothesisthat the subnormal A is cyclic. Itiseasytoseethatif Aand Barecyclicsubnormaloperatorssuchthat A B iscyclicsubnormal,then R(δ A,B )isorthogonalto ker(δ A,B ). Hence,itwouldbeinterestingtoestablishtherange-kernelorthogonalityof δ A,B in the general case. 2)Let π: L(H) L(H)/K(H) = C(H)denotetheCalkinmap,andlet S = {A L(H): π(a) = A }. NotethattheresultofDuggal[5]guaranteesthatif Aand Barecyclicsubnormal operators,then R(δ A,B )isorthogonalto ker(δ A,B ) S,and R( A,B )isorthogonal to ker( A,B ) S. From this, the following question naturally arises: If Aand Barecyclicsubnormaloperators,is R( A,B )orthogonalto ker( A,B ) for the usual operator norm? 3)Let A L(H),andsupposethat f isananalyticfunctiononanopenset containing σ(a)suchthat f doesnotvanishonsomeneighborhoodof σ(a). If f(a)isisometricornormal,whatconditionson fensuretherange-kernelorthogonalityof δ A withrespecttothewiderclassofunitarilyinvariantnormson L(H)? Acknowledgement. Itisourgreatpleasuretothanktherefereeforcareful reading of the paper and useful suggestions. 268

10 References [1] J. Anderson: On normal derivations. Proc. Am. Math. Soc. 38(1973), [2] C. A. Berger, B. I. Shaw: Selfcommutators of multicyclic hyponormal operators are always trace class. Bull. Am. Math. Soc. 79(1974), [3] S. Bouali, Y. Bouhafsi: On the range kernel orthogonality and P-symmetric operators. Math. Inequal. Appl. 9(2006), [4] M.B.Delai,S.Bouali,S.Cherki:Aremarkontheorthogonalityoftheimagetothekernel of a generalized derivation. Proc. Am. Math. Soc. 126(1998), (In French.) [5] B. P. Duggal: A perturbed elementary operator and range-kernel orthogonality. Proc. Am. Math. Soc. 134(2006), [6] B.P.Duggal: A remark on normal derivations. Proc. Am. Math. Soc. 126 (1998), [7] I. C. Gohberg, M. G. Kreĭn: Introduction to the Theory of Linear Nonselfadjoint Operators. Translations of Mathematical Monographs 18, American Mathematical Society, Providence, 1969; translated from the Russian, Nauka, Moskva, [8] F. Kittaneh: Normal derivations in norm ideals. Proc. Am. Math. Soc. 123 (1995), [9] Y. Tong: Kernels of generalized derivations. Acta Sci. Math. 54(1990), [10] A.Turnšek:Orthogonalityin C pclasses.monatsh.math.132(2001), [11] A. Turnšek: Elementary operators and orthogonality. Linear Algebra Appl. 317(2000), [12] T. Yoshino: Subnormal operator with a cyclic vector. Tôhoku Math. J. II. Ser. 21(1969), Authors addresses: Said Bouali, Department of Mathematics, Faculty of Science, Ibn Tofail University, B.P. 133, Kénitra, Morocco, said.bouali@yahoo.fr; Youssef Bouhafsi, Department of Mathematics, Faculty of Science, Chouaib Doukkali University, Iben Maachou Street, P.O.Box 20, El Jadida, Morocco, ybouhafsi@yahoo.fr. 269