ORTHOGONALITY IN S 1 AND S SPACES AND NORMAL DERIVATIONS

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1 J. OPERATOR THEORY 51(2004), c Copyrigh by Thea, 2004 ORTHOGONALITY IN S 1 AND S SPACES AND NORMAL DERIVATIONS DRAGOLJUB J. KEČKIĆ Communicaed by Florian-Horia Vasilescu Absrac. We inroduce ϕ-gaeaux derivaive, and use i o give he necessary and sufficien condiions for he operaor Y o be orhogonal (in he sense of James) o he operaor X, in boh spaces S 1 and S (nuclear and compac operaors on a Hilber space). Furher, we apply hese resuls o prove ha here exiss a normal derivaion A such ha ran A ker A S 1, and a relaed resul concerning S. Keywords: Gaeaux derivaive, orhogonaliy in Banach spaces, Schaen ideals, derivaion, elemenary operaor. MSC (2000): Primary 46G05, 47B10, 47B47; Secondary 47A30, 46B INTRODUCTION Le H denoe a separable Hilber space, and le S p denoe he Schaen ideal of ( + ) 1/p hose compac operaors X acing on H such ha X p = s (X) p < +, where s (X) = λ (X X) 1/2. Also, le S denoe he ideal of all compac operaors equipped wih he usual norm. Le us recall ha hese (Schaen) norms are special cases of so called uniarily invarian norms, associaed wih some wosided ideal of compac operaors. For furher deails he reader is referred o [6]. I is well known ha S 2 has a Hilber space srucure, wih he inner produc X, Y = r (XY ), and ha his is no rue in oher S p. Neverheless, in all Banach spaces we can define he orhogonaliy in he following way (orhogonaliy in he sense of R.C. James). =1

2 90 Dragolub J. Kečkić Definiion 0.1. Le X be a Banach space. We say ha y X is orhogonal o x X if for all complex numbers λ here holds (0.1) x + λy x. Remark 0.2. If X is a Hilber space, hen from (0.1) we can easily derive x, y = 0. Remark 0.3. In Banach spaces, orhogonaliy from he previous definiion is no symmerical, i.e. y orhogonal o x does no imply x orhogonal o y. Remark 0.4. Definiion 0.1 has a naural geomeric inerpreaion. Namely, y x if and only if he complex line {x + λy : λ C} is disoin wih he open ball K(0, x ), i.e. if and only if his complex line is a angen one. Such an orhogonaliy relaion is closely relaed wih Gaeaux derivaive of he norm and he smoohness of he sphere of radius x. Definiion 0.5. The vecor x is a smooh poin of he sphere S(0, x ) if here exiss a unique suppor funcional F x X, such ha F x (x) = x and F x = 1. Proposiion 0.6. If here exiss he Gaeaux derivaive of he norm a he poin x, i.e. if here exiss he i = 0, hen i is equal o R 0 x+y x Re F x (y), where F x is he funcional from he previous definiion. Moreover, in his case y is orhogonal o x if and only if F x (y) = 0. I is also well known ha if Banach space X has a sricly convex dual space hen every nonzero poin is a smooh poin of he corresponding sphere. For deails see [1] and references herein. Orhogonaliy in he sense of James were used in invesigaion of so called elemenary operaors, inroduced by Lumer and Rosenblum ([11]). Definiion 0.7. Le (A 1, A 2,..., A n ) and (B 1, B 2,..., B n ) be he n-uples of bounded Hilber space operaors. The mapping X o B(H) is called he elemenary operaor or elemenary mapping. n =1 A XB from B(H) Remark 0.8. The same name elemenary operaors is used for operaors of he same form, which maps J o J, where J is some wo sided ideal equipped wih a uniarily invarian norm. The firs resul concerning he orhogonaliy in he sense of James and elemenary operaors was given by Anderson ([2]).

3 Orhogonaliy in S 1 and S 91 Proposiion 0.9. If A is a normal operaor on a separable Hilber space H, hen AS = SA implies ha for all bounded X here holds AX XA + S S. In view of Definiion 0.1, i means ha he range of he mappings A : B(H) B(H), A (X) = AX XA is orhogonal o is kernel. This resul has been generalized in wo direcions, by exending he class of elemenary mappings, and by exending his inequaliy o he oher uniarily invarian norms; see for insance [4], [5], [8], [9]. In [2], Anderson also proved ha equaliy ran A ker A = B(H) is rue in very special cases, for example if and only if he specrum of he normal operaor A in Proposiion 0.9 is finie. In [8] here was conecured ha i migh be J = ran A J ker A J if he ideal J is separable. In Secion 3, we shall give he negaive answer o his hypohesis. The Gaeaux derivaive echnique was used in [3], [10] and [12], in order o characerize hose operaors o which he range of a derivaion is orhogonal. In hese papers, he aenion was direced o S p ideals for some p > 1, and o smooh poins in S 1 and S, like in he following proposiion, aken from [10]. Proposiion Le A be a bounded Hilber space operaor. The range of a derivaion A is orhogonal o an operaor S in S p if and only if A S = SA, where S = U S p 1, and S = U S. Smooh poins in S 1 and S, were characerized by Holub ([7]). Proposiion The operaor X is a smooh poin of he corresponding sphere in S 1 if and only if eiher X is inecive or X is inecive. The operaor X is a smooh poin of he corresponding sphere in S if and only if i aains is norm a he unique vecor (up o a complex scalar). The main purpose of his noe is o characerize he orhogonaliy in he sense of James in S 1 and S a he poins which are no smooh, and o apply hese characerizaions o elemenary operaors. Namely, among oher hings, we prove ha for a normal derivaion A : S p S p here holds ran A ker A = S p for 1 < p < +, and ha such compleeness resul fails for p = 1. In he case p = + he siuaion is more complicaed. Some resuls concerning more general elemenary operaors are also given.

4 92 Dragolub J. Kečkić 1. ϕ-gateaux DERIVATIVES In his secion we inroduce ϕ-gaeaux derivaive and, in Theorem 1.4, we give he necessary and sufficien condiion for a vecor y from an arbirary Banach space o be orhogonal (in he sense of James) o a vecor x, in erms of inroduced ϕ-gaeaux derivaive. Definiion 1.1. Le (X, ) be an arbirary Banach space. ϕ-gaeaux derivaive of he norm a he poin x, and in y-direcion is x + e iϕ y x D ϕ,x (y) =. 0 + Proposiion 1.2. (i) The funcion α x,y () = x + e iϕ y is convex. (ii) D ϕ,x (y) is he righ derivaive of he funcion α x,y a he poin 0, and aking ino accoun (i) D ϕ,x (y) always exiss. Proof. Obvious. Proposiion 1.3. (i) D ϕ,x is subaddiive, posiively homogeneous funcional on X; (ii) D ϕ,x (e iθ y) = D ϕ+θ,x (y); (iii) D ϕ,x (y) y. Proof. (i) We have x + e iϕ (y 1 + y 2 ) x 2 + eiϕ y 1 + x, 2 + eiϕ y 2 and, by aking a i we obain D ϕ,x (y 1 + y 2 ) = 0 + x + e iϕ (y 1 + y 2 ) x 0 + x + 2e iϕ y 1 + x + 2e iϕ y 2 2 x 2 = D ϕ,x (y 1 ) + D ϕ,x (y 2 ), which proves he subaddiiviy. Posiive homogeneiy is obvious. (ii) Obvious. (iii) I is enough o see ha x + e iϕ y x x + e iϕ y x = y. The previous simple consrucion allows us o characerize he orhogonaliy in he sense of James, in all Banach spaces (wihou care of smoohness) via ϕ- Gaeaux derivaive. Theorem 1.4. The vecor y is orhogonal o x in he sense of James if and only if inf ϕ D ϕ,x(y) 0. Proof. Le us firs prove he only if par of he saemen. Indeed, le y be orhogonal o x in he sense of James, i.e. le for all λ C here holds x + λy x. 0 for all > 0, and passing o he i we ge D ϕ,x (y) 0 for an arbirary ϕ. Then x+eiϕ y x

5 Orhogonaliy in S 1 and S 93 Le us, now, prove he oher, if, par of he saemen. We have D ϕ,x (e i(π ϕ) x + e iϕ e i(π ϕ) x x 1 1 x) = = x = x From his, and from subaddiiviy we ge x = D ϕ,x (e i(π ϕ) x) D ϕ,x (µy) D ϕ,x (e i(π ϕ) x) D ϕ,x (µy e i(π ϕ) x) µy e i(π ϕ) x = x + µ( e i(ϕ π) )y = x + λy, if we ake µ = e i(π ϕ) λ. Remark 1.5. We can see ha he previous heorem is reasonable if we look a i from an oher aspec. Namely, y is orhogonal o x if and only if he convex funcion α x,y () aains is minimum a he origin. We conclude his secion wih wo examples concerning wo classical Banach spaces. Example 1.6. In he space L 1 (X, µ) he funcion g is orhogonal o f, in he sense of James if and only if e iθ() g() dµ() g() dµ(), {f 0} {f=0} where f() = f() e iθ(). Indeed, in he L 1 space here holds { } D ϕ,f (g) = Re e iϕ e iθ() g() dµ() + g() dµ(), since {f 0} f() + ρe iϕ g() f() = ρ 0 + ρ (here g() = g() e iψ() ), and also f()+ρeiϕ g() f() ρ g f if and only if inf ϕ Re { {f 0} } e iϕ e iθ() g() dµ() + {f=0} { cos(ϕ θ() + ψ()) g(), f() 0, g(), f() = 0. g(). Thus, we ge {f=0} g() dµ() 0. However, he infimum will be aained for ha ϕ, for which e iϕ e iθ() g() dµ() = e iθ() g() dµ(), {f 0} and he resul follows. {f 0} Example 1.7. In he c 0 space, y is orhogonal o x if and only if here does no exis he open angle D = {z : α < arg z < β}, wih β α < π, such ha ξ ν η ν D for all hose ν for which ξ ν = x holds.

6 94 Dragolub J. Kečkić Le k 1, k 2,..., k n be all of indices, for which ξ k = x holds, and le δ > 0 be a real number such ha sup ξ ν = x δ, and le ξ ν = ξ ν e iθν. Now, for ν k < δ 2 y we have: { ξν + e iϕ η ν x δ 2 for ν = k ξ ν + e iϕ η ν x δ 2 for ν k. Thus x + e iϕ y = max 1 n ξ k + e iϕ η k, implying x + e iϕ y x D ϕ,x (y) = = x = max 1 n Re {eiϕ e iθ k ηk }, max 1 n 1 + eiϕ η k ξ k 1 aking ino accoun ha for all n-uples of complex numbers here holds max{ 1 + z 1, 1 + z 2,..., 1 + z n } 1 = max{re z 1, Re z 2,..., Re z n }. 0 + So, we have: y is orhogonal o x if and only if inf ϕ max ν=k Re e iϕ e iθ ν η ν ORTHOGONALITY IN S 1 AND S Theorem 2.1. Le X, Y S 1. Then, here holds 0 + X + Y S1 X S1 = Re {r (U Y )} + QY P S1, where X = U X is he polar decomposiion of he operaor X, P = P ker X, Q = P ker X. For he proof of his heorem we need hree echnical lemmas. Lemma 2.2. Le X = U X and X + Y = V X + Y be he polar decomposiions of he operaors X and X + Y, le Q(P ) be he proecor o he kernel of X (X), and le {χ } be some complee orhonormal sysem in ker X. Then: (i) V n x Ux, srongly, for all x ran X, and for some sequence n 0 +. Also, V n x U x, srongly, for all x ran X, and for some (or same) sequence n 0 +. (ii) n + V n (I Q)Y χ, χ = 0, provided ha Y is nuclear. Proof. (i) Le e be some complee orhonormal sysem in H. For each, he family {V e : > 0} is a bounded family, and hence here exiss a sequence n 0, such ha V n e converges weakly. Moreover, using Canor s diagonal process, we conclude ha here exiss a sequence n 0 such ha V n e converges weakly for all, and herefore V n converges weakly. Le V 0 denoe he weak i of he sequence V n. Now, for all y, z H we have V n X + n Y z, y =

7 Orhogonaliy in S 1 and S 95 (X + n Y )z, y. However, X + n Y converges srongly (even uniformly) o X, as well as X + n Y converges srongly o X, and passing o he i we ge V 0 X z, y = Xz, y = U X z, y for all z, y H. Thus V 0 x = Ux for all x ran X. Since ran X is dense in ran X, we obain ha V n converges weakly o Ux for all x ran X. However, his convergence is moreover srong. Indeed, le x be an arbirary vecor from ran X. Then here exiss z H such ha x = X z. We have ha V n X + n Y z ends weakly o Ux. Bu, V n X + n Y z = (X + n Y )z which ends srongly o Xz = Ux. Thus V n X + n Y z ends srongly o Ux. Now we have V n x Ux V n ( X z X + n Y z) + V n X + n Y z Xz, which ends o zero as n ends o infiniy. In a similar way, we can obain ha V n x ends o U x for all x ran X and for some (same) sequence n. (ii) By par (i) we have ha P V n (I Q)Y P converges srongly o P U (I Q)Y P, and by Theorem III.6.3. from [6] P V n (I Q)Y P ends o P U (I Q)Y P in nuclear norm, since Y is nuclear operaor. However, V n (I Q)Y χ, χ is precisely he race of he nuclear operaor P V n (I Q)Y P and herefore i ends o he race of he operaor P U (I Q)Y P. Bu P U = 0 and he proof is complee. Lemma 2.3. Le A be a bounded operaor, whose (usual) norm is a mos one, and le {ϕ } be an arbirary orhonormal sysem. Then, we have X 1 AXϕ, ϕ. Proof. Indeed AXϕ, ϕ r (AX) A X 1 X 1. Lemma 2.4. Le ϕ be some orhonormal sysem (no necessarily complee) in H. (i) for any vecor f H, and for all ε > 0, here exiss a vecor f, such ha f f < ε and f, ϕ < +. { (ii) he se F = A S 1 : } Aϕ < + is dense in S 1. Proof. (i) Le f = f 1 + f 2, f 1 L{ϕ }, f 2 L{ϕ }. Since here holds f 1 2 = f 1, ϕ 2, here exiss n 0, such ha f 1, ϕ 2 < ε 2. We define >n 0 f as f = f 1, ϕ ϕ + f 2. We have f, ϕ = f 1, ϕ < +, and n 0 n 0 also f f 2 = f 1, ϕ ϕ 2 = f 1, ϕ 2 < ε 2. >n 0 >n 0 k=1 k=1 (ii) Le Y S 1, and le Z = N σ k, f k g k (0 < σ k+1 σ k, f k, g k orhonormal sysems) be a finie rank operaor such ha Y Z 1 < ε 2. By he previous par of he saemen, here exis vecors f k, such ha f k, ϕ < +, and f k f k < ε 2 k σ k. Le A = N σ k, f k g k. We have A Z 1 = N σ k, f k k=1

8 96 Dragolub J. Kečkić f k g k 1 N k=1 k=1 σ k f k f k ε 2, and hence A Y 1 < ε. On he oher hand Aϕ N σ k ϕ, f k g k = N σ k ϕ, f k < +. k=1 Now, we are in a posiion o prove Theorem 2.1. Proof of Theorem 2.1. Le X = s, ϕ ψ be he Schmid expansion of he operaor X, and le χ be a complee orhonormal sysem in ker X. Then, aking ino accoun Lemma 2.3, we have: 1 { X + Y 1 X 1 } = 1 { X + Y 1 } s 1 ( U (X + Y )ϕ, ϕ + V (X + Y )χ, χ s ), where V : ker X ker X is given by QY P = V QY P. Furher 1 { X + Y 1 X 1 } 1 { X ϕ, ϕ + U Y ϕ, ϕ + V (X + Y )χ, χ = 1 { ( s + U Y ϕ, ϕ + ) V Y χ, χ s }. Bu, since V Q = V and P χ = χ he las expression is equal o 1 { X + Y 1 X 1 } = 1 { ( s + U Y ϕ, ϕ + ) V } QY P χ, χ s s + (r (U Y ) + QY P 1 ) s = Re (r (U Y ) + QY P 1 ), X+Y and hus 1 X Re (r (U Y )) + QY P 1. { We shall derive he opposie inequaliy for hose Y which belong o F = A S 1 : } Aϕ < +. I will be enough, since we will ge wo sublinear X+Y bounded funcionals Y 1 X and Y Re r (U Y ) + QY P 1, ha coincide on he se F which is dense by Lemma 2.4. Also, in he following, wherever V as 0 +, is wrien, i means V n as n +, where n is a sequence s }

9 Orhogonaliy in S 1 and S 97 from Lemma 2.2. (We do no need o care abou i, since by Proposiion 1.2 (ii) always exiss he i ha we consider.) A he firs we have 1 } { X + Y 1 X 1 (2.1) = 1 { X + Y ϕ, ϕ + X + Y χ, χ s }. However, 1 X + Y χ, χ = 1 V (X + QY )χ, χ + V (I Q)Y χ, χ, and also, QY P 1 V QY P χ, χ = V QY χ, χ = 1 V (X + QY )χ, χ, so ha a real number 1 X + Y χ, χ is equal o a sum of a complex number whose modulus is less or equal o QY P 1, and an oher complex number, whose modulus is, for small enough, less or equal o ε (Lemma 2.2). Thus, for small enough, we ge 1 X + Y χ, χ QY P 1 + ε. On he oher hand, by Jensen s inequaliy applied o he inegraion wih respec o he specral measure we have: X + Y ϕ, ϕ X + Y 2 ϕ, ϕ = = s 2 + 2Re Y ϕ, s ψ + 2 Y ϕ 2 and, applying (2.1) 1 { X + Y 1 X 1 } = s 2 + 2Re Y ϕ, s ψ + 2 Y ϕ 2 + QY P 1 + ε s 2 + 2Re Y ϕ, s ψ + 2 Y ϕ 2 s + QY P 1 + ε s = 2Re Y ϕ, s ψ + 2 Y ϕ 2 ) + QY P 1 + ε ( s 2 + 2Re Y ϕ, s ψ + 2 Y ϕ 2 + s Re Y ϕ, ψ + QY P 1 + ε = Re Y ϕ, Uϕ + QY P 1 + ε = Re (r U Y ) + QY P 1 + ε.

10 98 Dragolub J. Kečkić The inequaliy 2Re Y ϕ, s ψ + 2 Y ϕ 2 ) ( s 2 + 2Re Y ϕ 2Re Y ϕ, ψ + Y ϕ, s ψ + 2 Y ϕ 2 + s allows us o ake a i as 0 + under he sum. The resul now follows, since ε can be arbirarily small. The following corollary characerizes orhogonaliy in he sense of James in he space S 1. Corollary 2.5. The operaor Y is orhogonal o he operaor X in he space S 1 if and only if r (U Y ) QY P S1, where X = U X, P = P kerx, and Q = P kerx. Proof. By Theorem 1.4, Y is orhogonal o X if and only if inf D ϕ,x(y ) ϕ 0. However, by Theorem 2.1, here holds inf D ϕ,x(y ) = inf Re ϕ ϕ (eiϕ r (U Y )) + QY P 1, and we ge he resul, by choosing he mos suiable ϕ. Theorem 2.6. Le X, Y be in S. Then we have X + Y X 0 + = max Re U Y f, f, f Φ where X = U X, and Φ is he characerisic subspace of he operaor X wih respec o is eigenvalue s 1. For he proof of his heorem we need a echnical lemma, as well. Lemma 2.7. Le A be a posiive compac operaor, le Φ be he subspace where A aains is norm, le Φ γ be he se of hose vecors from he Hilber space H which forms wih Φ an angle less or equal o γ. Le, furher, B be a selfadoin compac operaor, such ha B δ, where δ is a real number such ha 2δ s 1(A) s 2(A) 2δ an γ. Then we have A + B = max (A + B)f, f. f Φ γ, Proof. If he uni vecor x is represened as x = f + g, where f Φ, g Φ, g hen x Φ γ if and only if f an γ. Also, s 2(A) = max Af, f. Since f Φ he operaor A + B is compac and selfadoin, here exiss a uni vecor x such ha (A + B)x, x = A + B. If we represen his vecor as x = f + g, hen i will be (A + B)x, x = (A + B)f, f + 2Re Bf, g + (A + B)g, g. Bu, since (A+B)f, f A+B f 2, Bf, g δ f g, (A+B)g, g (s 2 +δ) g 2, we have A + B A + B f 2 + 2δ f g + (s 2 + δ) g 2, i.e. (s 1 δ) g A + B g 2δ f + (s 2 + δ) g, aking ino accoun A + B s 1 δ, respecively (s 1 s 2 2δ) g 2δ f, from which we conclude x Φ γ.

11 Orhogonaliy in S 1 and S 99 Proof of Theorem 2.6. A firs, we have X + Y X (X + Y )(X + Y ) 1/2 X = X X + (X Y + Y X) + 2 Y Y X 2 = 0 + ( X X + (X Y + Y X) + 2 Y Y 1/2 + X ). I is obvious ha he denominaor X X + (X Y + Y X) + 2 Y Y 1/2 + X ends o 2 X. Le us consider he i of he numeraor. The operaors X X and (X Y +Y X)+ 2 Y Y saisfy he assumpions of Lemma 2.7, for an arbirary γ > 0 and for he corresponding small enough, and we ge X X + (X Y + Y X) + 2 Y Y s = 0 + max f Φ γ (X X + (X Y + Y X) + 2 Y Y )f, f s 2 1 max [ (X Y + Y X)f, f + Y Y f, f ] = max 2Re Y f, Xf. 0 + f Φ γ f Φ γ On he oher hand X X + (X Y + Y X) + 2 Y Y s = 0 + max (X X + (X Y + Y X) + 2 Y Y )f, f s 2 1 f Φ max f Φ so ha for all γ > 0 we have [ (X Y + Y X)f, f + Y Y f, f ] = max 2Re Y f, Xf, f Φ 1 X max X + Y X Re Y f, Xf 1 f Φ γ 0 + X max f Φ γ Noe ha inf max γ>0 f Φ γ Re Y f, Xf = max Re Y f, Xf, f Φ Re Y f, Xf. since Y and X are coninuous in he sphere meric. So, we can ge he resul, by aking an infimum over all γ, since for f Φ here holds Xf = X Uf. Corollary 2.8. In he space S he following hree condiions are muually equivalen: (i) Y is orhogonal o X in he sense of James. (ii) inf 0 ϕ<2π max f Φ Re e iϕ U Y f, f 0, where X = U X and Φ is he subspace where he operaor X aains is norm.

12 100 Dragolub J. Kečkić (iii) There exiss he vecor f Φ such ha Y f Xf. Proof. The equivalence beween (i) and (ii) follows from Theorems 1.4 and 2.6. However, he condiion (ii) ells us ha he numerical range of he operaor U Y (on he subspace Φ) has in he complex plane, such a posiion ha i conains a leas one value wih posiive real par, under all roaions around he zero, i.e. ha is no conained in an open half-plane, whose boundary conains he origin. Bu by Toepliz-Haussdorf Theorem he numerical range is a closed convex se, so he las condiion is equivalen o he condiion ha he numerical range of he operaor U Y conains he origin. Since he vecors Uf and Xf always have he same direcion, we conclude ha (iii) is equivalen o (ii). 3. THE SUM OF THE RANGE AND THE KERNEL OF THE ELEMENTARY OPERATORS Le us, firs, recall some facs concerning ideals of compac operaors. Proposiion 3.1. If J is a separable ideal of compac operaors, associaed wih some uniarily invarian norm, hen is dual is isomerically isomorphic wih anoher ideal of compac operaors (no necessarily separable) and i admis he represenaion: ϕ Y (X) = r (XY ). Proof. This is, in fac, Theorem III from [6]. Proposiion 3.2. Le J be some separable ideal of compac operaors, and le E : J J be some elemenary operaor given by E(X) = n A XB. Then is conugae operaor E : J J has he form E (Y ) = n B Y A. Proof. We have ( n ) ϕ Y (E(X)) = r (E(X)Y ) = r A XB Y = r ( X =1 =1 =1 n ) B Y A = r (XE (Y )) = ϕ E (Y )(X). =1 Consider an arbirary separable ideal of compac operaors J, such ha J is sricly convex. According o Proposiion 0.9, for all X J here exiss a unique operaor X J such ha X(X) = X and X = 1. If, moreover, J is reflexive hen he mapping X X, X = ω(x) is a biecion (and also involuion) of he uni spheres of he spaces J and J. Moreover, Y is orhogonal o X in he space J if and only if X(Y ) = 0.

13 Orhogonaliy in S 1 and S 101 Theorem 3.3. Le J be a reflexive ideal in B(H) such ha J is sricly convex, and le E : J J be an elemenary operaor given by E(X) = n A XB. Then ran E is orhogonal (in he sense of James) o he operaor S if and only if ω(s) = S ker E. Proof. Taking ino accoun Proposiions 0.6 and 3.2, we have ha ran E S implies ha for all X J, S(E(X)) = 0 or (E ( S))(X) = 0, for all X, and consequenly E ( S) = 0. Remark 3.4. Theorem 3.3 is he general resul and i holds on an arbirary Banach space. Lema 3.5. Le X be a reflexive Banach space and le V be a closed subspace of X. If V = {x X : v V v + x x } = {0} hen V = X. Proof. This is Lemma 3.6. from [14]. Theorem 3.6. Le J saisfy he assumpions of he previous heorem, and le E : J J be an elemenary operaor given by E(X) = AXB + CXD, where A, B, C and D are normal operaor such ha AC = CA, BD = DB and A A + C C > 0, B B + D D > 0. Then J = ran E ker E. Proof. In [8], i is proved ha for such elemenary operaors is range is orhogonal o is kernel, and by his and by previous Theorem we have he following implicaions: E(S) = 0 X J, E(X) + S S E ( S) = 0 =1 X J, E (X) + S S E ( S) = 0 E(S) = 0. Thus we have E(S) = 0 if and only if E(X) S for all X J. From he orhogonaliy of he range and he kernel i follows ha he sum ran E + ker E is closed. Indeed, if x n + y n for x n ran E, y n ker E ends o z, hen, by inequaliy y n y m x n + y n x m y m we conclude ha y n is a Cauchy sequence, and herefore y n y ker E. Furher x n z y ran E, and hus z ran E + ker E. Suppose ha E(X) + Y + Z J Z J, for all X, and for all Y ker E. By choosing Y = 0 we see ha Z ker E. Now, we can pu Y = Z and X = 0, implying Z = 0. Hence (ran E + ker E) = {0}. This, by Lemma 3.5, finishes he proof. Corollary 3.7. Le p > 1, and le E : S p S p, E(X) = AXB + CXD, where A, B, C and D are as in he previous heorem. Then S p = ran E ker E. Moreover, for any elemenary operaor on S p i is valid ha ran E is orhogonal o S if and only if E ( S p 1 U ) = 0. Proof. I is well known ha S p = S q, q > 1, and ha S q is sricly convex (Clarckson-McCarhy inequaliies; see [13]). Furher, we can easily check ha in he case of S p, S = 1 S p 1 U, which concludes he proof. S p/q p

14 102 Dragolub J. Kečkić Remark 3.8. The special case of his heorem is Proposiion 3 from [10]. Theorem 3.9. There exiss a normal derivaion A : S 1 S 1, A (X) = AX XA, wih AA = A A such ha S 1 ran A ker A. Proof. Le H = l 2 (Z), and le A be he bilaeral shif operaor, i.e. for all n Z le Ae n = e n 1. Le us, firs, perceive ha he kernel of he derivaion A is rivial. Indeed, le X ker A. Then AX = XA, implies Xe i, e = Xe i, A e 1 = AXe i, e 1 = XAe i, e 1 = Xe i 1, e 1. However, aking ino accoun he compacness of he operaor X we ge 0 = Xe i+n, e +n = Xe i, e, for n + all i, Z, and herefore X = 0. Thus ran A ker A = ran A. Le us now consruc he operaor S S 1 in he following way: { Se = 0 for 0, Se = 1 2 e 1 for > 0. If S = U S hen clearly U e = 0 for < 0, and U e = e +1 for 0. We shall prove ha ran A is orhogonal o S. Indeed, his orhogonaliy is, by Corollary 2.5, equivalen o r (U (AX XA)) Q(AX XA)P 1, where P = P ker S and Q = P ker S. However r (U (AX XA)) = r ((AU U A)X), whereas AU U A =, e 0 e 0, and, in fac r (U (AX XA)) = Xe 0, e 0. On he oher hand i is easy o check ha P = P, Q = P L(...,e 2,e 1,e 0) L(...,e, 2,e 1) and we ge (aking in Lemma 2.3 he bounded operaor A ) Q(AX XB)P S1 + Q(AX XA)P e +1, e = = = 1 = = Xe 0, e 0, finishing he proof. + = ( AXe +1, e XAe +1, e ) = (AX XA)P e +1, Qe 1 = ( Xe +1, e +1 Xe, e ) Remark The rivial kernel of he derivaion from he previous heorem is a compleely unessenial deail. Indeed, considering Hilber space H H and he operaor A I on i, we can consruc a normal derivaion which has he same properies as ha in Theorem 3.9, and whose kernel is nonrivial. Theorem There exiss a normal derivaion B : S S, B (X) = BX XB, wih BB = B B, and an operaor S S such ha ran B is orhogonal o S, and S / ker B. Proof. Le A be he operaor from he proof of Theorem 3.9, and le B = A I acing on H = l 2 (Z) C. Furher, le S = S 1 2I, where S 1 : l 2 (Z) l 2 (Z) is any operaor of norm a mos one. The operaor S aains is norm a he unique (up o a scalar) vecor ϕ = 0 1 l 2 (Z) C. I is obvious ha Sϕ = 2ϕ, Bϕ = B ϕ = ϕ, and herefore (BX XB)ϕ, Sϕ = 2 Xϕ, B ϕ 2 XBϕ, ϕ = 2 Xϕ, ϕ 2 Xϕ, ϕ = 0.

15 Orhogonaliy in S 1 and S 103 Thus ran B S. On he oher hand BS = SB implies AS 1 = S 1 A, and S 1 = 0. So if we ake S 1 0 we are done. Remark I is no possible o prove S ran B ker B. On he conrary, he sum ran B ker B is always equal o S. Namely, le f Y S be a funcional of he form f Y (X) = r (XY ) for some Y S 1 = S which annihilaes ran B. I immediaely follows BY Y B = 0, i.e. Y ker B. Since f Y (Y ) 0, f Y can no annihilae ker B. Thus ran B + ker B is always dense in S. Remark One can find ha Remark 3.12 is in a confusion wih Theorem However, his is a consequence of he fac ha boh V = {X S : U V, X + U U } and V = {X S : U V, X + U X }, in general, does no make a subspace, bu a cone! Acknowledgemens. The auhor is deeply graeful o Professor V.S. Shulman for suggesions in formulaing Theorem 3.11, and o Professor D.R. Jocić for suggesions in proving Theorem 2.1. REFERENCES 1. T.J. Abazoglu, Norm derivaives on spaces of operaors, Mah. Ann. 239(1979), J. Anderson, On normal derivaions, Proc. Amer. Mah. Soc. 38(1973), S. Bouali, S. Cherki, Approximaion by generalized commuaors, Aca Sci. Mah. (Szeged) 63(1997), B.P. Duggal, A remark on normal derivaions, Proc. Amer. Mah. Soc. 126(1998), B.P. Duggal, Range-kernel orhogonaliy of he elemenary operaors X np A ixb i X, Linear Algebra Appl. 337(2001), i=1 6. I.C. Gohberg, M.G. Kreĭn, Inroducion o he Theory of Linear Nonselfadoin Operaors, Transl. Mah. Monogr., vol. 18, Amer. Mah. Soc., Providence, RI, J.R. Holub, On he meric geomery of ideals of operaors on Hilber space, Mah. Ann. 201(1973), D. Kečkić, Orhogonaliy of he range and he kernel of some elemenary operaors, Proc. Amer. Mah. Soc. 128(2000), F. Kianeh, Normal derivaions in norm ideals, Proc. Amer. Mah. Soc. 123(1995), F. Kianeh, Operaors ha are orhogonal o he range of a derivaion, J. Mah. Anal. Appl. 203(1996), G. Lumer, M. Rosenblum, Linear operaor equaions, Proc. Amer. Mah. Soc. 10(1959), P.J. Maher, Commuaor approximans, Proc. Amer. Mah. Soc. 115(1992), B. Simon, Trace Ideals and heir Applicaions, London Mah. Soc. Lecure Noe Ser., vol. 35, Cambridge Univ. Press, Cambridge 1979.

16 104 Dragolub J. Kečkić 14. A. Turnšek, Orhogonaliy in C p classes, Monash. Mah. 132(2001), DRAGOLJUB J. KEČKIĆ Faculy of Mahemaics Universiy of Belgrade Sudenski rg Beograd YUGOSLAVIA keckic@poincare.maf.bg.ac.yu Received November 13, 2001; revised Sepember 15, 2002 and February 22, 2003.