In this paper, we minimize the map Fp (X)= ||S−(AX−XB)||Pp ,
where the pair (A, B) has the property (F P )Cp , S ∈ Cp , X varies such that
AX − XB ∈ Cp and Cp denotes the von Neumann-Schatten class

Mecheri, Salah

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica Math. J. 26 (2000), 119-126ON MINIMIZING ‖S − (AX −XB)‖ppSalah MecheriCommunicated by S. L. TroyanskiAbstract. In this paper, we minimize the map Fp(X)= ‖S−(AX−XB)‖pp,where the pair (A,B) has the property (FP )Cp , S ∈ Cp, X varies such thatAX −XB ∈ Cp and Cp denotes the von Neumann-Schatten class.1. Introduction. Let L(H) be the algebra of all bounded operatorsacting on a complex Hilbert space H. For A and B in L(H), let δA,B denote theoperator on L(H) defined by δA,B(X) = AX −XB. If A = B, then δA is calledthe inner derivation induced by A. A well-known result of J. Anderson and C.Foias [1] says that if A and B are normal operators such that, AS = SB then,for all X ∈ L(H),‖S − (AX −XB)‖ ≥ ‖S‖ .(1.1)In this paper we obtain an inequality similar to (1.1), where the operator norm isreplaced by the ‖ · ‖p norm on the von Neumann-Schatten class Cp, 1 ≤ p <∞.We prove that, if the pair (A,B) has the property (FP )Cp , i.e. (AT = TB, where2000 Mathematics Subject Classification: Primary 47B47, 47A30; Secondary 47B10.Key words: Generalized Derivation, Orthogonality, von Neumann-Schatten class, PutnamFuglede′s property.120 Salah MecheriT ∈ Cp implies A∗T = TB∗) 1 ≤ p <∞ and S ∈ ker δA,B ∩Cp then, the map Fpdefined by Fp(X) = ‖S − (AX −XB)‖pp has a global minimizer at V if, and for1 < p <∞ only if, AV − V B = 0. In other words, we have‖S − (AX −XB)‖pp ≥ ‖T‖pp(1.2)if, and for 1 < p < ∞ only if, AV − V B = 0. Thus in Halmos′ terminologie [5]the zero commutator is the commutator approximant in Cp of T . Additionally,we show that if, the pair (A,B) has the property (FP )Cp and S ∈ ker δA,B ∩Cp,1<p<∞ then, the map Fp has a critical point atW if, and only if, AW−WB = 0,i.e. if DWFp is the Frechet derivative at W of Fp, the set{W ∈ L(H) : DWFp = 0}(1.3)coinsides with ker δA,B (the kernel of δA,B).2. Preliminaries. For details of the von Neumann-Schatten class see [8].Theorem 2.1 [2]. If 1 < p <∞, then the map Fp : Cp 7−→ R+ definedby X 7−→ ‖X‖pp, is differentiable at every X ∈ Cp with derivative DXFp given byDXFp(T ) = pRe tr(|X|p−1 U∗T ),(2.1)where tr denotes trace, Re z is the real part of a complex number z and X = U |X|is the polar decomposition of X. If dimH < ∞, then the same result holds for0 < p ≤ 1 at every invertible X.Theorem 2.2 [6]. If U is a convex set of Cp, with 1 < p <∞, then themap X 7−→ ‖X‖pp, where X ∈ U has at most a global minimizer.3. Orthogonality. The following definition generalizes the idea of or-thogonality in Hilbert space.Definition 3.1. Let C be a complex numbers and let E be a normedlinear space. Let F and G be two subspaces of E. If ‖x+ y‖ ≥ ‖y‖ for all x ∈ Fand for all y ∈ G, then F is said to be orthogonal to G.Definition 3.2. Let A,B ∈ L(H). The pair (A,B) has the property(FP )L(H) if, AC = CB, where C ∈ L(H) implies A∗C = CB∗.Definition 3.3. Let U(A,B) = {X ∈ L(H) : AX −XB ∈ Cp} andFp : U 7−→ R+ be the map defined by Fp(X) = ‖T − (AX −XB)‖pp, whereT ∈ ker δA,B ∩ Cp, 1 ≤ p <∞.Minimization 121Theorem 3.1 [7]. Let A ∈ L(H), if A is normal and S ∈ Cp, 1 ≤ p <∞such that AS = SA, then‖S − (AX −XA)‖pp ≥ ‖S‖pp ,(3.1)for every X ∈ L(H).Theorem 3.2. Let A,B ∈ L(H), if A and B are normal operators andT ∈ Cp, 1 ≤ p <∞ such that AT = TB, then‖T − (AX −XB)‖pp ≥ ‖T‖pp ,(3.2)for every X ∈ L(H).P r o o f. Taking on H⊕H,Q =[A 00 B], S =[0 T0 0], Y =[0 X0 0],then Q is normal on H ⊕ H and S ∈ Cp. Since AT = TB, then it results thatQS = SQ. SinceQY − Y Q =[0 AX −XB0 0],then it follows from Teorem 3.1 that,‖S − (QY − Y Q)‖pp ≥ ‖S‖pp , ∀S ∈ Cpconsequently we obtain,‖T − (AX −XB)‖pp = ‖S − (QY − Y Q)‖pp ≥ ‖S‖pp = ‖T‖pp . Lemma 3.3. Let A,B ∈ L(H). The following statements are equivalent:(1) The pair (A,B) has the property (FP )L(H)(2) If AT = TB where T ∈ L(H), then R(T ) reduces A, ker(T )⊥ reducesB, and A |R(T ) and B |ker(T )⊥ are normal operators, where R and ker denote therange and the kernel, respectively.P r o o f. (1) ⇒ (2): Since AT = TB and the pair (A,B) has the property(FP )L(H), A∗T = TB∗ and so R(T ) and ker(T )⊥ are reducing subspaces for Aand B, respectively. Since A(AT ) = (AT )B, we obtain A∗(AT ) = (AT )B∗ by(FP )L(H), and the identity A∗T = TB∗ implies that A∗AT = AA∗T . Thus we seethat A |R(T )is normal. Clearly (B∗, A∗) satisfies (FP )L(H), and B∗T ∗ = T ∗A∗.122 Salah MecheriTherefore it follows from the above argument that B∗ |R(T ∗)= (B |ker(T )⊥)∗ isnormal.(2) ⇒ (1): Let T ∈ L(H) such that AT = TB. Taking the two decom-positions of H, H1 = H = R(T )⊕R(T )⊥, H2 = H = ker T ⊕ ker(T )⊥. Then wecan write A and B on H1 into H2 respectively,A =[A1 00 A2], B =[B1 00 B2],where A1, B1 are normal operators. Also we can write T and X on H2 into H1T =[T1 00 0],X =[X1 X2X3 X4].It follows from AT = TB that A1T1 = T1B1. Since A1 and B1 are normaloperators, then by applying the Putnam Fuglede′s theorem, we obtain A∗1T1 =T1B∗1 , that is, A∗T = TB∗. Theorem 3.4. Let A,B ∈ L(H). If the pair (A,B) has the property(FP )L(H), then‖C − (AX −XB)‖ ≥ ‖C‖ ,(3.3)for every operator C ∈ ker δA,B and for every X ∈ L(H).P r o o f. Since the pair (A,B) has the property (FP )L(H), it follows fromLemma 3.3 that, R(C) reduces A, ker(C)⊥ reduces B, and A |R(C)and B |ker(C)⊥are normal operators. Let,A =[A1 00 A2], B =[B1 00 B2], C =[C1 00 0], X =[X1 X2X3 X4].It follows fromAC − CB =[A1C1 − C1B1 00 0]= 0,that A1C1 = C1A1 and we have‖C − (AX −XB)‖ =∥∥∥∥[C1 − (A1X1 −X1B1) ∗∗ ∗]∥∥∥∥ .Since A1 and B1 are two normal operators, then the result of J. H. An-derson and C. Foias [1] guarentees that,‖C1 − (A1X1 −X1B1)‖ ≥ ‖C1‖ ,Minimization 123so‖C +AX −XB‖ ≥ ‖C1 − (A1X1 −X1B1)‖ ≥ ‖C1‖ = ‖C‖ . Remark 3.1. If A and B are two normal operators, then inequality(3.3) holds for every C ∈ ker δA,B. Hence, Theorem 3.4 generalizes the resultgiven by J. H. Anderson and C. Foias [1]. In particular we haveR(δA,B) ∩ ker δA,B = {0} .(3.4)Corollary 3.5. Let A,B ∈ L(H) and C ∈ ker δA,B, then‖C +AX −XB‖ ≥ ‖C‖ ,∀X ∈ L(H).In each of the following cases:(1) A dominant and B∗ M−hyponormal(2) A dominant and B∗ k−quasihyponormal(3) A k−quasihyponormal and B∗ k−quasihyponormal injective(4) A k−quasihyponormal and B∗ dominant.P r o o f. Adapted from B. P. Duggal [3] if we have (1), (2), (3)and (4) thepair (A,B) has the property (FP )L(H). Lemma 3.6. Let A,B ∈ L(H) and C ∈ L(H) such that the pair(A,B) has the property (FP )L(H). If A |S|p−1 U∗ = |S|p−1 U∗B, where p > 1 andS = U |S| is the polar decomposition of S, then A |S|U∗ = |S|U∗B.P r o o f. If T = |S|p−1, thenATU∗ = TU∗B.(3.5)We prove thatAT nU∗ = T nU∗B,(3.6)for all n ≥ 1. If S = U |S|, then kerU = ker |S| = ker |S|p−1 = kerT and(kerU)⊥ = (ker T )⊥ = R(T ). This shows that the projection U∗U onto (ker T )⊥satifies U∗UT = T and TU∗UT = T 2. By taking adjoints of (3.5) and sincethe pair (A,B) has the property (FP )L(H), we get BUT = UTA and AT2 =ATU∗UT = TU∗BUT = TU∗UTA = T 2A. Since A commutes with the positiveoperator T 2, then A commutes with its square roots, that is,AT = TA(3.7)124 Salah MecheriBy (3.5) and (3.7) we obtain (3.6). Let f(t) be the map defined on σ(T ) ⊂ R+by f(t) = t1p−1 ; 1 < p < ∞. Since f is a uniform limits of a sequence (Pi) ofpolynomials without constant term (since f(0) = 0), then it follows from (3.6)that APi(T )U∗ = Pi(T )U∗B. Therefore AT1p−1U∗ = U∗T1p−1B. Theorem 3.7. Let A,B ∈ L(H). If the pair (A,B) has the property(FP )Cp and S ∈ Cp such that AS = SB, then1) For 1 ≤ p < ∞, the map Fp has a global minimizer at W if, and for1 < p <∞ only if, AW −WB = 0.2) For 1 < p < ∞, the map Fp has a critical point at W if, and only if,AW −WB = 0.3) For 0 < p ≤ 1, dimH < ∞ and S − (AW −WB) is invertible, thenFp has a critical point at W , if AW −WB = 0.Before proving this theorem we need the following lemma.Lemma 3.8. Let A,B ∈ L(H). The following statements are equivalent.(1) The pair (A,B) has the property (FP )Cp(2) If AT = TB where T ∈ Cp, then R(T ) reduces A, ker(T )⊥ reduces B,and A |R(T )and B |ker(T )⊥ are normal operators.P r o o f. Since Cp is a bilateral ideal and T ∈ Cp, then AT ∈ Cp. It sufficesto remark that A(AT ) = (AT )B implies A∗(AT ) = (AT )B∗ by (FP )Cp , and theidentity A∗T = TB∗ implies that A∗AT = AA∗T . By the same arguments as inthe proof of Lemma 3.3, the proof of this Lemma can be finished. P r o o f o f T h e o r em 3.8. Since the pair (A,B) has the property(FP )Cp , it follows from the above lemma that, R(S) reduces A, ker(S)⊥ reducesB, and A |R(S) and B |ker(S)⊥ are normal operators. Let,A =[A1 00 A2], B =[B1 00 B2], S =[S1 00 0], X =[X1 X2X3 X4].It follows fromAS − SB =[A1S1 − S1B1 00 0]= 0,that A1S1 = S1B1 and we have‖S − (AX −XB)‖pp =∥∥∥∥[S1 − (A1X1 −X1B1) ∗∗ ∗]∥∥∥∥pp.Minimization 125Since A1 and B1 are two normal operators, then it results from Theo-rem 3.2 that,‖S1 − (A1X1 −X1B1)‖pp ≥ ‖S1‖pp ,hence it follows from [4, p. 82] that,‖S − (AX −XB)‖pp ≥ ‖S1 − (A1X1 −X1B1)‖pp ≥ ‖S1‖pp = ‖S‖pp ,i.e. Fp(X) ≥ Fp(W ). Conversely, if Fp has a minimum then ‖S − (AW −WB)‖pp =‖S‖pp. Since U is convex then, the set V = {S − (AX −XB); X ∈ U} is also con-vex. Thus Theorem 2.2 implies that S − (AW −WB) = S.2) Let W,S ∈ U and φ, ϕ be two maps defined respectively byφ : X 7−→ S − (AX −XB); ϕ : X 7−→ ‖X‖pp .Since the Frechet derivative of Fp is given byDWFp(T ) = limh−→0Fp(W + hT )− Fp(W )h,it follows that DWFp(T ) = [DS−(AW−WB)](TB−AT ). If W is a critical point ofFp, then DWFp(T ) = 0, ∀T ∈ U , by applyig Theorem 2.1 we get,DWFp(T ) = pRe tr[|S − (AW −WB)|p−1W ∗(TB −AT ) =pRe tr[Y (TB −AT )] = 0,where S − (AW −WB) = W |S − (AW −WB)| is the polar decomposition ofthe operator S − (AW − WB) and Y = |S − (AW −WB)|p−1W ∗. An easycalculation shows that BY − Y A = 0, that is,A |S − (AW −WB)|p−1W ∗ = |S − (AW −WB)|p−1W ∗B.It follows from Lemma 3.6 thatA |S − (AW −WB)|W ∗ = |S − (AW −WB)|W ∗B.By taking adjoints and since the pair (A,B) has the property (FP )Cp , we getA(T−(AW−WB)) = (T−(AW−WB))B, then A(AW−WB) = (AW−WB)B.Hence AW −WB ∈ R(δA,B) ∩ ker δA,B , by applying the equality(3.4) it resultsthat AW −WB = 0.Conversely, if AW = WB, then W is a minimum and since Fp is differ-entible, then W is a critical point.126 Salah Mecheri3) Suppose that dimH < ∞. If AW − WB = 0, then S is invertibleby hypothesis. Also |S| is invertible, hence |S|p−1 exists for 0 < p ≤ 1. TakingY = |S|p−1U∗, where S = U |S| is the polar decomposition of S. Since AS = SBimplies S∗A = BS∗, then S∗AS = BS∗S and this implies that |S|2B = B|S|2 and|S|B = B|S|. Since S∗A = BS∗, i.e. |S|U∗A = B|S|U∗, then |S|(U∗A−BU∗) =0 and since B|S|p−1 = |S|p−1B, so BY − Y A = B|S|p−1U∗ − |S|p−1U∗A =|S|p−1(BU∗ − U∗A). So, that BY − Y A = 0 and tr[(BY − Y A)T ] = 0 for everyT ∈ L(H). Since S = S−(AW−WB), that is 0 = tr[Y TB−Y AT ] = tr[Y (TB−AT )] = pRe tr[Y (TB − AT )] = pRe tr[|S|p−1 U∗(TB − AT )] = (DTφ)(TB −AT ) = (DWFp)(T ). Remark 3.2. In Theorem 3.7 the implication W is a critical point ⇒AW − WB = 0, does not hold in the case 0 < p ≤ 1 because the functionalcalculus argument involving the function t 7−→ t1p−1 , where 0 ≤ t < ∞, is onlyvalid for 1 < p <∞.REFERENCES[1] J. H. Anderson, C. Foias. Properties which normal operator share withnormal derivation and related operators. Pacific. J. Math. (1975), 313–325.[2] A. H. Almoadjil. The commutants of relatively prime powers in Banachalgebra. Proc. Amer. Math. Soc. 57 (1976), 243–251.[3] B. P. Duggal. On generalized Putnam-Fuglede theorem. Monatshe. Math.107 (1989), 309–332.[4] I. C. Gohberg, M. G. Krein. Introduction to the theory of linear non-selfadjoint operators. Transl. Math. Monographs vol. 18, Amer. Math. Soc.,Providence, 99–110.[5] P. R. Halmos. Positive approximants of operators. Indiana Univ. Math. J.21 (1972), 99–110.[6] P. J. Maher. Some norm inequalities concerning generalized inverses. LinearAlgebra Appl. 174 (1992), 951–960.[7] P. J. Maher. Commutator approximant. Proc. Amer. Math. Soc. 115(1992), 995–1000.[8] J. R. Ringrose. Compact Nonselfadjoint Operators. Van Nostrand, Rhein-hold, London, 1971University of Tebessa12000 Tebessa, Algeriae-mail: mecherisalah@hotmail.com Received February 21, 2000

On Minimizing ||S−(AX−XB)||Pp

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On Minimizing ||S−(AX−XB)||Pp

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