Isometries of perfect norm ideals of compact operators
aa r X i v : . [ m a t h . OA ] M a r ISOMETRIES OF PERFECT NORM IDEALS OF COMPACTOPERATORS
BEHZOD AMINOV AND VLADIMIR CHILIN
Abstract.
It is proved that for every surjective linear isometry V on a perfectBanach symmetric ideal C E = C of compact operators, acting in a complexseparable infinite-dimensional Hilbert H there exist unitary operators u and v on H such that V ( x ) = uxv or V ( x ) = ux t v for all x ∈ C E , where x t is atranspose of an operator x with respect to a fixed orthonormal basis for H . Inaddition, it is shown that any surjective 2-local isometry on a perfect Banachsymmetric ideal C E = C is a linear isometry on C E . Introduction
Let H be a complex separable infinite-dimensional Hilbert space. Let ( E, k·k E ) ⊂ c be a real Banach symmetric sequence space. Consider an ideal C E of compactlinear operators in H , which is defined by the relations x ∈ C E ⇐⇒ { s n ( x ) } ∞ n =1 ∈ E and k x k C E = k{ s n ( x ) } ∞ n =1 k E , where { s n ( x ) } ∞ n =1 are the singular values of x (i.e. the eigenvalues of ( x ∗ x ) / indecreasing order). In the paper [10] it is shown that k · k C E is a Banach norm on C E . In addition, C := C l ⊂ C E . In the case when ( E, k · k E ) is a separable Banach space, the Banach ideal( C E , k · k C E ) is a minimal Banach ideal in the terminology of Schatten [16], i.e. theset of finite rank operators is dense in C E .It is known [18] that for every surjective linear isometry V on a minimal Banachideal C E = C , where C := C l , there exist unitary operators u and v on H suchthat(1) V ( x ) = uxv (or V ( x ) = ux t v )for all x ∈ C E , where x t is the transpose of the operator x with respect to afixed orthonormal basis in H . In the case of a Banach ideal C , a description ofsurjective linear isometries of the form of (1) was obtained in [15], and for the ideals C p := C l p , ≤ p ≤ ∞ , p = 2, in [2].In this paper we show that, in the case a Banach symmetric sequence space E = l with Fatou property, every surjective linear isometry V on C E has the form(1). In addition, it is proved that in this case any 2-local surjective isometry on C E also is of the form (1). Date : March 4, 2017.2010
Mathematics Subject Classification.
Key words and phrases.
Banach symmetric ideal of compact operators, Fatou property, Her-mitian operator, 2-local isometry. Preliminaries
Let l ∞ (respectively, c ) be a Banach lattice of all bounded (respectively, con-verging to zero) sequences { ξ n } ∞ n =1 of real numbers with respect to the norm k{ ξ n }k ∞ = sup n ∈ N | ξ n | , where N is the set of natural numbers. If x = { ξ n } ∞ n =1 ∈ l ∞ ,then a non-increasing rearrangement of x is defined by x ∗ = { ξ ∗ n } , where ξ ∗ n := inf | F | 0) whenever a k ∈ E (respectively, a k ∈ C E ) and a k ↓ 0. It is known that( C E , k·k C E )) has order continuous norm if and only if ( E, k·k E ) has order continuousnorm (see, for example, [5, Proposition 3.6]). Moreover, every Banach symmetricideal with order continuous norm is a minimal norm ideal, i.e. the subspace F ( H )is dense in C E .If ( E, k · k E ) is a Banach symmetric sequence space (respectively, ( C E , k · k C E ) isa Banach symmetric ideal), then the K¨othe dual E × (respectively, C × E ) is definedas E × = { ξ = { ξ n } ∞ n =1 ∈ l ∞ : ξη ∈ l for all η ∈ E } (respectively , C × E = { x ∈ B ( H ) : xy ∈ C for all y ∈ C E } )and k ξ k E × = sup { ∞ X n =1 | ξ n η n | : η = { η n } ∞ n =1 ∈ E, k η k E ≤ } if ξ ∈ E × SOMETRIES OF PERFECT NORM IDEALS OF COMPACT OPERATORS 3 (respectively , k x k C × E = sup { tr ( | xy | ) : y ∈ C E , k y k C E ≤ } if x ∈ C × E ) , where tr ( · ) is the standard trace on B ( H ).It is known that ( E × , k · k E × ) (respectively, ( C × E , k · k C × E ) is a Banach symmetricsequence space (respectively, a Banach symmetric ideal). In addition, C × = B ( H )and if C E = C , then C × E ⊂ K ( H ) [7, Proposition 7]. We also note the followinguseful property [5, Theorem 5.6]:( C × E , k · k C × E ) = ( C E × , k · k C E × ) . A Banach symmetric ideal C E is said to be perfect if C E = C ×× E . It is clear that C E is perfect if and only if E = E ×× .A Banach symmetric sequence space ( E, k·k E ) (respectively, a Banach symmetricideal ( C E , k · k C E )) is said to possess Fatou property if the conditions0 ≤ a k ≤ a k +1 , a k ∈ E (respectively , a k ∈ C E ) for all k ∈ N and sup k ≥ k a k k E < ∞ (respectively , sup k ≥ k a k k C E < ∞ )imply that there exists a ∈ E ( a ∈ C E ) such that a k ↑ a and k a k E = sup k ≥ k a k k E (respectively, k a k C E = sup k ≥ k a k k C E ).It is known that ( E, k · k E ) (respectively, ( C E , k · k C E )) has the Fatou propertyif and only if E = E ×× [12, Vol. II, Ch. 1, Section a] (respectively, C E = C ×× E [5,Theorem 5.14]). Therefore C E is perfect ⇔ C E = C ×× E ⇔ E = E ×× ⇔ E has the Fatou property ⇔ C E has Fatou property . If y ∈ C × E , then a linear functional f y ( x ) = tr ( xy ) = tr ( yx ) , x ∈ C E , is continuouson C E . In addition, k f y k C ∗ E = k y k C × E , where C ∗ E is the dual of the Banach space( C E , k · k C E ) (see, for example, [7]). Identifying an element y ∈ C × E and the linearfunctional f y , we may assume that C × E is a closed linear subspace in C ∗ E . Since F ( H ) ⊂ C × E , it follows that C × E is a total subspace in C ∗ E . Thus, the weak topology σ ( C E , C × E ) is a Hausdorff topology. Proposition 1. [4, Proposition 2.8] A linear functional f ∈ C ∗ E is continuous withrespect to weak topology σ ( C E , C × E ) (i.e. f ∈ C × E ) if and only if f ( x n ) → for everysequence x n ↓ , x n ∈ C E , n ∈ N . It is clear that C hE = { x ∈ C E : x = x ∗ } is a real Banach space with respect tothe norm k · k C E ; besides, the cone C + E = { x ∈ C hE : x ≥ } is closed in C hE and C hE = C + E − C + E . If 0 ≤ x n ≤ x n +1 ≤ x, x, x n ∈ C hE , n ∈ N , then there existssup n ≥ x n ∈ C hE , i.e. C hE is a quasi-( o )-complete space [1]. Hence, by [1, Theorem1], we have the following proposition. Proposition 2. Every linear functional f ∈ ( C hE ) ∗ is a difference of two positivelinear functionals from ( C hE ) ∗ . We need the following property of the weak topology σ ( C E , C × E ). Proposition 3. If x ∈ C E , then there exists a sequence { x n } ⊂ F ( H ) such that x n σ ( C E , C × E ) −→ x . BEHZOD AMINOV AND VLADIMIR CHILIN Proof. It suffices to establish the validity of Proposition 3 for0 ≤ x = ∞ X j =1 λ j p j ∈ C E \ F ( H ) , where λ j ≥ x and p j ∈ F ( H ) arefinite-dimensional projectors for all j ∈ N (the series converges with respect tothe norm k · k ∞ ). If y n = ∞ P j = n λ j p j , then y n ↓ f ( y n ) → f ∈ C × E . Therefore F ( H ) ∋ x − y n σ ( C E , C × E ) −→ x. (cid:3) Let T be a bounded linear operator acting in a Banach symmetric ideal ( C E , k ·k C E ), and let T ∗ be its adjoint operator. If T ∗ ( C × E ) ⊂ C × E , then f ( T ( x α )) = T ∗ ( f )( x α ) → T ∗ ( f )( x ) = f ( T ( x ))for every net x α σ ( C E , C × E ) −→ x, x α , x ∈ C E , and for all f ∈ C × E . Thus T is a σ ( C E , C × E )-continuous operator.Inversely, if the operator T is a σ ( C E , C × E )-continuous and f ∈ C × E , then the linearfunctional ( T ∗ f )( x ) = f ( T ( x )) is also σ ( C E , C × E )-continuous, i.e. T ∗ ( f ) ∈ C × E .Therefore, a linear bounded operator T : C E → C E is σ ( C E , C × E )-continuous ifand only if T ∗ ( f ) ∈ C × E for every f ∈ C × E . Thus, for any λ ∈ C and linear boundedoperators T, S : C E → C E , continuous with respect to a σ ( C E , C × E )-topology, theoperators T + S , T S and λT are σ ( C E , C × E )-continuous.Let B ( C E ) be the Banach space of bounded linear operators T acting in ( C E , k ·k C E ) with the norm k T k B ( C E ) = sup k x k C E ≤ k T ( x ) k C E . Proposition 4. If T n ∈ B ( C E ) are σ ( C E , C × E ) -continuous operators, T ∈ B ( C E ) ,and k T n − T k B ( C E ) → as n → ∞ , then T is also σ ( C E , C × E ) -continuous.Proof. It suffices to show that T ∗ ( f ) ∈ C × E for any functional f ∈ C × E . Since T n ∈ B ( C E ) are σ ( C E , C × E )-continuous operators, it follows that T ∗ n ( f ) ∈ C × E forall f ∈ C × E , n ∈ N . Considering the closed subspace C × E in ( C ∗ E , k · k C ∗ E ) and notingthat k T ∗ n ( f ) − T ∗ ( f ) k C ∗ E ≤ k T ∗ n − T ∗ k B ( C ∗ E ) k f k C ∗ E = k T n − T k B ( C E ) k f k C ∗ E → , we conclude that T ∗ ( f ) ∈ C × E . (cid:3) We also need the following well-known properties of perfect symmetrically normedideals. Theorem 1. Let ( C E , k · k C E ) ⊂ c be a Banach symmetric sequence space withFatou property. Then ( i ) . [5, Theorem 5.11] . k x k C E = sup y ∈C × E , k y k C× E ≤ | tr ( xy ) | for every x ∈ C E ; SOMETRIES OF PERFECT NORM IDEALS OF COMPACT OPERATORS 5 ( ii ) . [4, Theorem 3.5] . A Banach space C E is σ ( C E , C × E ) -sequentially complete,i.e. if for x n ∈ C E , n ∈ N , and for every f ∈ C × E there is lim n →∞ f ( x n ) , then thereexists x ∈ C E such that x n σ ( C E , C × E ) −→ x . The ball topology in norm ideals of compact operators Let ( X, k · k X ) be a real Banach space and let b X be the ball topology in X , i.e. b X is the coarsest topology such that every closed ball B ( x, ε ) = { y ∈ X : k y − x k X ≤ ε } , ε > , is closed in b X [8]. The family X \ n [ i =1 B ( x i , ε i ) , x i ∈ X, k x − x k X > ε i , i = 1 , ..., n, n ∈ N . is a base of neighborhoods of the point x ∈ X in b X . Therefore x α b X −→ x, x α , x ∈ X , if and only if lim inf k x α − x k X ≥ k x − y k X for all y ∈ X [8]. In particular, everysurjective isometry V in ( X, k · k X ) is continuous with respect to the ball topology b X .Let us note that b X is not a Hausdorff topology. The following theorem provides asufficient condition for T -axiom of b X on subsets of X . Recall that A is a Rosenthalsubset of ( X, k · k X ) if every sequence in A has a weakly Cauchy subsequence. Theorem 2. [8, Theorem 3.3] . Let ( X, k · k X ) be a real Banach space, and let A be a bounded absolutely convex Rosenthal subset of X . Then ( A, b X ) is a Hausdorffspace. In the proof of required properties of the ball topology (see Theorem 3 below),we utilize the following well-known proposition. Proposition 5. [14, section 3, Ch. 1, § . Let p nk be real numbers, n, k ∈ N , suchthat n P k =1 | p nk | = 1 for all n ∈ N , and let the limit lim n →∞ p nk = p k exist for every fixed k ∈ N . Then the sequence s n = p n r + p n r + . . . + p nn r n converges for every convergent sequence { r n } . Theorem 3. Let ( C E , k · k C E ) be a Banach symmetric ideal, x n ∈ C + E , and let x n ↓ . Then the sequence { x n } can converge with respect to the topology b C E to nomore than one element.Proof. Consider C E as a real Banach space. Let A be the absolutely convex hull ofthe sequence { , x , ..., x n , ... } . Since x n ↓ k · k C E is monotone, itfollows that A is a bounded subset of C E . Let us show that A is a Rosenthal subset.Using the inequality 0 ≤ x n +1 ≤ x n and a decomposition of linear functional f ∈ ( C hE ) ∗ as the difference of two positive functionals from ( C hE ) ∗ (see Proposition2), we conclude that there exists lim n →∞ f ( x n ) = r n for every functional f ∈ ( C hE ) ∗ .For any sequence { y n } ∞ n =1 ⊂ A , we have y n = p n x + p n x + . . . + p nk ( n ) x k ( n ) , where k ( n ) X i =1 | p ni | = 1; BEHZOD AMINOV AND VLADIMIR CHILIN in particular, p ni ∈ [ − , 1] for all i ∈ , . . . , k ( n ). Consider a sequence q n = ( p n , p n . . . , p nk ( n ) , , . . . ) ∈ ∞ Y i =1 [ − , . By Tychonoff Theorem, a set ∞ Q i =1 [ − , 1] is compact with respect to the producttopology. Moreover, by [11, ch. 4, theorem 17], the set ∞ Q i =1 [ − , 1] is a metrizablecompact. Hence, the sequence { q n } has a convergent subsequence { q n i } ∞ i =1 . Inparticular, there are the limits lim n i →∞ p n i k for all k ∈ N . Besides, the sequence f ( y n i ) = k P j =1 p n i j f ( x j ) , f ∈ ( C hE ) ∗ , satisfies all conditions of Proposition 5, whichimplies its convergence. Therefore, the sequence { y n } ∞ n =1 ⊂ A has a weakly Cauchysubsequence { y n i } . This means that A is a Rosenthal subset of ( C hE , k · k C E ).By Theorem 2, the topological space ( A, b C hE ) is Hausdorff, hence the sequence { x n } ∞ n =1 ⊂ A could not have more than one limit with respect to the topology b C hE .Since C hE is a closed subspace in ( C E , k · k C E ) (we assume that C E is a real space),it follows that the restriction ( b C E ) | C hE is finer than b C hE . Therefore, the sequence { x n } ∞ n =1 can have no more than one limit with respect to the topology b C E . (cid:3) Proposition 6. If E ⊂ c is a Banach symmetric sequence space with Fatouproperty, then b C E ≤ σ ( C E , C × E ) .Proof. It suffices to show that B (0 , 1) = { x ∈ C E : k x k C E ≤ } is closed withrespect to the weak topology σ ( C E , C × E ). Let x α ∈ B (0 , 1) and x α σ ( C E , C × E ) −→ x ∈ C E .Assume that x / ∈ B (0 , k x k C E = q > 1, and let ǫ > q − ǫ > i ), there exists y ∈ C × E such that k y k C × E ≤ q ≥ | tr ( xy ) | > q − ǫ .On the other hand, x n σ ( C E , C × E ) −→ x implies that | tr ( x n y ) | → | tr ( xy ) | . Since x n ∈ B (0 , | tr ( x n y ) | ≤ | tr ( xy ) | ≤ 1, which isimpossible. Thus B (0 , 1) is a closed set in σ ( C E , C × E ). (cid:3) Weak continuity of Hermitian operators in the perfect ideals ofcompact operators In this section, we establish σ ( C E , C × E )-continuity of the Hermitian operator act-ing in a perfect Banach symmetric ideal C E of compact operators. For that we need σ ( C E , C × E )-continuity of a surjective isometry on C E .Recall that the series ∞ P n =1 x n converges weakly unconditionally in a Banach space X if a numerical series ∞ P n =1 f ( x n ) converges absolutely for every f ∈ X ∗ [19, Ch. 2, § Proposition 7. [19, Ch. 2, § . Let ( X, k · k X ) be a Banach space, x n ∈ X, n ∈ N .Then the following conditions are equivalent: ( i ) A series ∞ P n =1 x n converges weakly unconditionally; SOMETRIES OF PERFECT NORM IDEALS OF COMPACT OPERATORS 7 ( ii ) There exists a constant C > , such that sup N k N X n =1 t n x n k X ≤ C k{ t n } ∞ n =1 k ∞ for all { t n } ∞ n =1 ∈ l ∞ . Proposition 7 implies the following. Corollary 4. If V is a surjective linear isometry on a Banach space X and aseries ∞ P n =1 x n converges weakly unconditionally in X , then the series ∞ P n =1 V ( x n ) alsoconverges weakly unconditionally in X . Now we can show that every surjective linear isometry of C E is σ ( C E , C × E )-continuous. Proposition 8. Let C E be a perfect Banach symmetric ideal of compact operatorsand V a surjective linear isometry on C E . Then V is σ ( C E , C × E ) -continuous.Proof. It suffices to show that V ∗ ( f ) ∈ C × E for each functional f ∈ C × E . Accordingto Proposition 1, it should be established that x n ↓ , { x n } ∞ n =1 ⊂ C + E , implies f ( V ( x n )) = V ∗ ( f )( x n ) → { x n } ∞ n =1 ⊂ C E and x n ↓ 0. We will show that the series ∞ P n =1 ( x n − x n +1 )converges weakly unconditionally in C E . If f ∈ ( C hE ) ∗ is a positive linear functional,then m X n =1 | f ( x n − x n +1 ) | = f ( m X n =1 ( x n − x n +1 )) = f ( x − x m +1 ) ≤ f ( x )for all m ∈ N . Hence the numerical series ∞ P n =1 f ( x n − x n +1 ) converges absolutely.Since every functional f ∈ ( C hE ) ∗ is the difference of two positive functionals from( C hE ) ∗ (see Proposition 2), the series ∞ P n =1 ( x n − x n +1 ) converges weakly uncondition-ally in C hE .Let f ∈ C ∗ E . Denote u ( x ) = Ref ( x ) = f ( x ) + f ( x )2 , v ( x ) = Imf ( x ) = f ( x ) − f ( x )2 i , x ∈ C E . It is clear that u, v ∈ ( C hE ) ∗ . Therefore, the series ∞ P n =1 u ( x n − x n +1 ) and ∞ P n =1 v ( x n − x n +1 ) converge absolutely. Thus, the series ∞ P n =1 f ( x n − x n +1 ) also converges abso-lutely. This means that a series ∞ P n =1 ( x n − x n +1 ) converges weakly unconditionallyin C E .By Corollary 4, the series ∞ X n =1 ( V x n − V x n +1 ) BEHZOD AMINOV AND VLADIMIR CHILIN converges weakly unconditionally in C E . Hence the numerical series ∞ X n =1 f ( V x n − V x n +1 ) = f ( V x ) − lim n →∞ f ( V x n )converges for every f ∈ C ∗ E . In particular, the limit lim n →∞ f ( V x n ) exists. Since C E is a σ ( C E , C × E )-sequentially complete set (see Theorem 1( ii )) and V is a bijection,it follows that there exists x ∈ C E such that V ( x n ) σ ( C E , C × E ) −→ V ( x ) as n → ∞ .It remains to show that V ( x ) = 0. By Proposition 6, V ( x n ) b C E −→ V ( x ) . Since the isometry V − is continuous with respect to the topology b C E , it followsthat x n b C E −→ x . Now, taking into account that x n ↓ 0, we obtain x n σ ( C E , C × E ) −→ x n b C E −→ 0, and Theorem 3 implies that x = 0. (cid:3) Let ( X, k·k X ) be a complex Banach space. A bounded linear operator T : X → X is called Hermitian, if the operator e itT = ∞ P n =0 ( itT ) n n ! is an isometry of the space X for all t ∈ R (see, for example, [6, Ch. 5, § C E is σ ( C E , C × E )-continuous. Theorem 5. Let E ⊂ c be a Banach symmetric sequence space with Fatou prop-erty, and let T be a Hermitian operator acting in C E . Then T is σ ( C E , C × E ) -continuous.Proof. Consider the non-negative continuous function α ( t ) = k e itT − I k B ( C E ) , where t ∈ R . As α (0) = 0, it follows that there exists t ∈ R such that α ( t ) < 1. Sincethe operator T is Hermitian, it follows that the operator V = e it T (and hence V − = e − it T ) is an isometry on C E . By Proposition 8, S = V − I is a σ ( C E , C × E )-continuous operator. In addition, k S k = α ( t ) < 1. Now, since it T = ln ( I + S ) = ∞ X n =1 ( − n − S n n , Proposition 4 implies that it T is a σ ( C E , C × E )-continuous operator. Therefore, T is a σ ( C E , C × E )-continuous operator. (cid:3) If a, b are self-adjoint operators in B ( H ), x ∈ C E , and(2) T ( x ) = ax + xb for all x ∈ C E , then T is a Hermitian operator acting in C E [18]. In the followingtheorem, utilizing the method of proof of Theorem 1 in [18], we show that everyHermitian operators acting in a perfect norm ideal ( C E , k · k C E ) = C has the form(2). Theorem 6. Let E ⊂ c be a Banach symmetric sequence space with Fatou prop-erty, E = l , and let T be a Hermitian operator acting in a Banach symmetricideal C E . Then there are self-adjoint operators a, b ∈ B ( H ) such that T ( x ) = ax + xb for all x ∈ C E . SOMETRIES OF PERFECT NORM IDEALS OF COMPACT OPERATORS 9 Proof. As in the proof of Theorem 1 from [18], we have that there are self-adjointoperators a, b ∈ B ( H ) such that T ( x ) = ax + xb for all x ∈ F ( H ). Fix x ∈ C E . ByProposition 3, there is a sequence { x n } ⊂ F ( H ) such that x n σ ( C E , C × E ) −→ x . If y ∈ C × E ,then tr ( y ( ax n + x n b )) = tr (( ya ) x n ) + tr (( by ) x n ) → tr (( ya ) x ) + tr (( by ) x ) == tr ( y ( ax ) + tr ( y ( xb ))) = tr ( y ( ax + xb )) . Therefore ax n + x n b σ ( C E , C × E ) −→ ax + xb. Since T is a Hermitian operator, it follows that T is σ ( C E , C × E )-continuous (seeTheorem 5). Therefore T ( x ) = σ ( C E , C × E ) − lim n →∞ T ( x n ) = σ ( C E , C × E ) − lim n →∞ ( ax n + x n b ) = ax + xb. (cid:3) Isometries of a Banach symmetric ideal In this section, we prove our main result, Theorem 7. The proof of Theorem 7 issimilar to the proof of Theorem 2 in [18]. We use a version of Theorem 1 in [18] fora perfect Banach symmetric ideal C E (Theorem 6) as well as σ ( C E , C × E )-continuityof every isometry on C E (Proposition 8) and σ ( C E , C × E )-density of the space F ( H )in C E (Proposition 3).Recall that x t stands for the transpose of an operator x ∈ K ( H ) with respect toa fixed orthonormal basis in H . Theorem 7. Let E ⊂ c be a Banach symmetric sequence space with Fatou prop-erty, E = l , and let V be a surjective linear isometry on the Banach symmetricideal C E . Then there are unitary operators u and v on H such that (3) V ( x ) = uxv ( or V ( x ) = ux t v ) for all x ∈ C E Note that each linear operator of the form (3) is an isometry on every Banachsymmetric ideal C E . Proof. Let y ∈ B ( H ), and let l y ( x ) = yx (respectively, r y ( x ) = xy ) for all x ∈ C E .It is clear that l y and r y are bounded linear operators acting in C E . UsingTheorem 6 and repeating the proof of the Theorem 2 [18], we conclude that thereare unitary operators u, v ∈ B ( H ) such that V l y V − = l uyu ∗ and V r y V − = r v ∗ yv for any y ∈ B ( H ).In the case V l y V − = l uyu ∗ , we define an isometry V on C E by the equation V ( x ) = u ∗ V ( x ) v ∗ . As in the proof of Theorem 2 [18], we get that V ( x ) = λx forevery x ∈ F ( H ) and some λ ∈ C . Since the isometry V is σ ( C E , C × E )-continuous(Proposition 8) and the space F ( H ) is σ ( C E , C × E )- dense in C E (Proposition 3), itfollows that V ( x ) = λx for every x ∈ C E . Now, since V is an isometry on C E , wehave λ = 1, i.e. V ( x ) = uxv for all x ∈ C E .In the case V l y V − = r v ∗ yv , as in the proof of Theorem 2 [18], we use the aboveto derive that there are unitary operators u, v ∈ B ( H ) such that V ( x ) = ux t v forall x ∈ C E . (cid:3) Corollary 8. Let E ⊂ c be a Banach symmetric sequence space with Fatou prop-erty, E = l , and let V be a surjective linear isometry on C E . Then V ( yx ∗ y ) = V ( y )( V ( x )) ∗ V ( y ) for all x, y ∈ C E .Proof. By Theorem 7, there are unitary operators u and v on H such that V ( x ) = uxv or V ( x ) = ux t v for all x ∈ C E . If V ( x ) = uxv , then V ( y )( V ( x )) ∗ V ( y ) = ( uyv )( uxv ) ∗ ( uyv ) = uyx ∗ yv = V ( yx ∗ y ) , x, y ∈ C E . Since ( x t ) ∗ = ( x ∗ ) t , x ∈ B ( H ), in the case V ( x ) = ux t v , we have V ( y )( V ( x )) ∗ V ( y ) = ( uy t v )( ux t v ) ∗ ( uy t v ) == uy t ( x ∗ ) t y t v = u ( y ( x ∗ ) y ) t v = V ( yx ∗ y ) , x, y ∈ C E . (cid:3) Let ( X, k · k X ) be an arbitrary complex Banach space. A surjective (not nec-essarily linear) mapping T : X → X is called a surjective 2-local isometry [13],if for any x, y ∈ X there exists a surjective linear isometry V x,y on X such that T ( x ) = V x,y ( x ) and T ( y ) = V x,y ( y ). It is clear that every surjective linear isometryon X is automatically a surjective 2-local isometry on X . In addition, T ( λx ) = V x,λx ( λx ) = λV x,λx ( x ) = λT ( x )for any x ∈ X and λ ∈ C . In particular,(4) T (0) = 0 . Thus, in order to establish linearity of a 2-local isometry T , it is sufficient to showthat T ( x + y ) = T ( x ) + T ( y ) for all x, y ∈ X .Note also that(5) k T ( x ) − T ( y ) k X = k V x,y ( x ) − V x,y ( y ) k X = k x − y k X for any x, y ∈ X . Therefore, in the case a real Banach space X , from (4), (5) andMazur-Ulam Theorem (see, for example, [6, Chapter I, § X is a linear. For complex Banachspaces, this fact is not valid.Using the description of isometries on a minimal Banach symmetric ideal C E from [18], L. Molnar proved that every surjective 2-local isometry on a minimalBanach symmetric ideal C E is necessarily linear [13, Corallary 5].The following Theorem is a version of Molnar’s result for a perfect Banach sym-metric ideal. Theorem 9. Let E ⊂ c be a Banach symmetric sequence space with Fatou prop-erty, E = l , and let V be a surjective 2-local isometry on a Banach symmetricideal C E . Then V is a linear isometry on C E .Proof. Fix x, y ∈ F ( H ) and let V x,y : C E → C E be a surjective isometry such that V ( x ) = V x,y ( x ) and V ( y ) = V x,y ( y ). By Theorem 7, there are unitary operators u and v on H such that V ( x ) = uxv or V ( x ) = ux t v (respectively, V ( y ) = uyv or V ( y ) = uy t v ). Then we have tr ( V ( x )( V ( y )) ∗ ) = tr ( V x,y ( x )( V x,y ( y )) ∗ ) = tr ( xy ∗ )for every x, y ∈ F ( H ). In addition, V ( x ) ∈ F ( H ) and V is a bijective mapping on F ( H ). SOMETRIES OF PERFECT NORM IDEALS OF COMPACT OPERATORS 11 If x, y, z ∈ F ( H ), then tr ( V ( x + y )( V ( z )) ∗ ) = tr (( x + y ) z ∗ ) , tr ( V ( x ) V ( z ) ∗ ) = tr ( xz ∗ ) ,tr ( V ( y ) V ( z ) ∗ ) = tr ( yz ∗ ) . Therefore tr (( V ( x + y ) − V ( x ) − V ( y ))( V ( z )) ∗ ) = 0for all z ∈ F ( H ). Taking z = x + y , z = x and z = y , we obtain tr (( V ( x + y ) − V ( x ) − V ( y )(( V ( x + y ) − V ( x ) − V ( y )) ∗ ) = 0 , that is, V ( x + y ) = V ( x ) + V ( y ) for all x, y ∈ F ( H ). Hence V is a bijective linearisometry on a normed space ( F ( H ) , k · k C E ).Let I be the closure of the subspace F ( H ) in the Banach space ( C E , k · k C E ). It isclear that ( I , k · k C E ) is a minimal Banach symmetric ideal. Since V is a surjectivelinear isometry on ( F ( H ) , k · k C E ), it follows that V is a surjective linear isometryon ( I , k · k C E ).By Theorem 2 [18], there are unitary operators u and v on H such that V ( x ) = u xv or V ( x ) = u x t v for all x ∈ I . Repeating the ending of the proof ofTheorem 1 in [13], we conclude that V is a linear isometry on C E . (cid:3) References [1] T. Ando. On fundamental properties of Banach space with a cone. Pacific J. Math. 12(1962),1163-1169[2] J. Arazy. The isometries of C p . Israel J. Math. 22(1975), 247-256.[3] J. 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