Ball proximinality of M -ideals of compact operators
aa r X i v : . [ m a t h . F A ] A ug Ball proximinality of M -ideals of compact operators C. R. Jayanarayanan and Sreejith Siju
Abstract.
In this article, we prove the proximinality of closed unit ball of M -ideals of compact operators.We also prove the ball proximinality of M -embedded spaces in their biduals. Moreover, we show that K ( ℓ ), the space of compact operators on ℓ , is ball proximinal in B ( ℓ ), the space of bounded operatorson ℓ , even though K ( ℓ ) is not an M -ideal in B ( ℓ ).
1. IntroductionOne of the important problems which arises naturally in approximation theory is the problem ofexistence of best approximation for a given vector from a specified subset of a Banach space. If thishappens for every vector, then the subset is said to be proximinal. Clearly compact subsets of a Banachspace are proximinal. However, lack of sufficient compact sets in infinite dimensional Banach spaces makesthe existence problem non-trivial. The present article studies the proximinality of closed unit ball of spaceof compact operators when it is an M -ideal in the space of bounded linear operators.Throughout this article, X will denote an infinite dimensional Banach space and H will denote aninfinite dimensional Hilbert space. For a Banach space X ; let B X , S X and B ( x, r ) denote the closed unitball, the unit sphere and the closed ball with center at x ∈ X and radius r respectively. We considerevery Banach space X , under the canonical embedding, as a subspace of X ∗∗ . For a Banach space X , let B ( X ) denote the space of all bounded linear operators on X , and K ( X ) denote the space of all compactoperators on X . The essential norm k T k e of an operator T ∈ B ( X ) is the distance from T to the spaceof compact operators, that is, k T k e = d ( T, K ( X )).We first recall some basic definitions and results which will be needed later. Definition 1.1.
Let K be a non-empty closed subset of a Banach space X . For x ∈ X , let P K ( x ) = { k ∈ K : d ( x, K ) = k x − k k} , where d ( x, K ) denotes the distance of x from K . An element of P K ( x ) is calleda best approximation to x from K. The set K is said to be proximinal in X if P K ( x ) = φ for all x ∈ X .A closed subspace Y of X is said to be ball proximinal in X if B Y is proximinal in X (see [3] for details).Clearly, ball proximinal subspaces are proximinal but the converse need not be true (see [3, Proposi-tion 2.4] and [10, Theorem 1]).In [5], Blatter raised the problem of identifying those Banach spaces X which are proximinal in itsbidual X ∗∗ , a problem which gained a lot of interest since the appearance of [5]. In general, a Banach Mathematics Subject Classification.
Primary 47L05,41A50 secondary 47B07, 47A58, 46B20.
Key words and phrases. M -ideals, Compact operators, M -embedded spaces, Ball proximinality. space need not be proximinal in its bidual (see [5, 7] for examples). In [7], Holmes and Kripke provedthat for a Hilbert space H , K ( H ) is proximinal in its bidual B ( H ). They have also asked whether K ( X )is proximinal in B ( X ) when X is a Banach space. In general, K ( X ) need not be proximinal in B ( X ).In the present paper, we discuss the ball proximinality of K ( X ) in B ( X ). Since K ( X ) may not be evenproximinal, we cannot expect ball proximinality of K ( X ) in general. So we discuss the ball proximinalityof K ( X ) under some additional assumption on X . Towards this we recall the notion of an M -ideal whichis stronger than proximinality. Definition 1.2 ([6]) . Let X be a Banach space and Y be a closed subspace of X . A linear projection P on X is said to be an L -projection ( M -projection ) if k x k = k P x k + k x − P x k ( k x k = max {k P x k , k x − P x k} )for all x ∈ X . A closed subspace Y of X is said to be an M -ideal in X if annihilator of Y in X ∗ ,denoted by Y ⊥ , is the range of an L -projection in X ∗ . If X is an M -ideal in X ∗∗ , then X is said to bean M -embedded space.It is well known that K ( H ) is an M -ideal in B ( H ) and K ( ℓ p ) is an M -ideal in B ( ℓ p ) for 1 < p < ∞ (see[6]). Hence K ( H ) and K ( ℓ p ) (1 < p < ∞ ) are proximinal in their respective biduals. By [6, Chapter VI,Proposition 4.11], we know that if X is reflexive and K ( X ) is an M -ideal in B ( X ), then K ( X ) ∗∗ = B ( X )and hence K ( X ) is proximinal in its bidual. However, M -ideals need not be ball proximinal (see [8]). Soit is natural to ask whether an M -embedded space is ball proximinal in its bidual. In this article, we givean affirmative answer to this question and also prove the ball proximinality of K ( X ) when K ( X ) is an M -ideal in B ( X ).In Section 2, we give an operator theoretic proof of ball proximinality of K ( H ) in B ( H ). We also provethe distance formula, d ( T, B K ( H ) ) = max {k T k − , d ( T, K ( H )) } for T ∈ B ( H ) . Another approach to study proximinality of space of compact operators is by using the following notionof basic inequality, introduced by Axler, Berg, Jewell, and Shields (see [1, 2] for details). Recall that a net( A α ) in B ( X ) converges to A ∈ B ( X ) in strong operator topology (SOT) if k A α x − Ax k −→ x ∈ X , and ( A α ) converges to A in weak operator topology (WOT) if x ∗ ( A α x ) −→ x ∈ X and x ∗ ∈ X ∗ . Definition 1.3 ([2]) . A Banach space X is said to satisfy the basic inequality if for each T ∈ B ( X ) andeach bounded net ( A α ) in K ( X ) such that A α → A ∗ α → ε >
0, there exists an index β such that k T + A β k ≤ ε + max ( k T k , k T k e + k A β k ) . The basic inequality was originally defined in [1] where sequences were used instead of nets (see Section 2of [2] to see the consequence of replacing sequences by nets in the definition of basic inequality).In Section 3, we prove the ball proximinality of K ( X ) in B ( X ) when K ( X ) is an M -ideal in B ( X ). Thekey idea for proving this result is the inequality mentioned in Lemma 3.2. This inequality is basically amodification of the revised basic inequality discussed in [13]. In fact, inequality in Lemma 3.2 and revisedbasic inequality in [13] are modified versions of the basic inequality. We also prove the ball proximinality ALL PROXIMINALITY OF M -IDEALS OF COMPACT OPERATORS 3 of M -embedded spaces in their biduals. Moreover, we show that K ( ℓ ) is ball proximinal in B ( ℓ ) eventhough K ( ℓ ) is not an M -ideal in B ( ℓ ).2. Ball proximinality of space of compact operators on Hilbert space.In this section, we prove the ball proximinality of K ( H ) in B ( H ) when H is a Hilbert space.Let S ( H ) be the set of all sequences of unit vectors in the Hilbert space H which converge weakly to0. For T ∈ B ( H ), define ∆( T ) = sup { lim sup n →∞ k T s n k : ( s n ) ∈ S ( H ) } as in [7]. In [7, Section 3, Theorem],it is proved that d ( T, K ( H )) = ∆( T ). The following theorem derives an analogous distance formula for B K ( H ) and establishes the ball proximinality of K ( H ) in B ( H ). Theorem 2.1.
Let H be an infinite dimensional Hilbert space. Then (1) d ( T, B K ( H ) ) = max {k T k − , ∆( T ) } = max {k T k − , d ( T, K ( H )) } for T ∈ B ( H ) and (2) K ( H ) is ball proximinal in B ( H ) .Proof. Let T ∈ B ( H ). Since, by [7], d ( T, K ( H )) = ∆( T ); it follows that max {k T k − , ∆( T ) } ≤ d ( T, B K ( H ) ). Without loss of generality, we may assume that T is not a compact operator. For, if T is a compact operator then T k T k is a best approximation to T from B K ( H ) . Case 1 : k T k > T does not attain its norm (that is, k T k 6 = k T x k for every x ∈ S H ). Let ( s n ) be a sequencein S H such that lim n →∞ k T s n k = k T k . Then, by [7, Lemma 2], ( s n ) ∈ S ( H ). Hence k T k ≤ ∆( T ) ≤ d ( T, B K ( H ) ). Thus 0 is a best approximation to T from B K ( H ) .Now let T ∈ B ( H ) be a norm attaining operator (that is, there exists an x ∈ S H such that k T k = k T x k ).We follow a technique similar to the one used in Case 3 of [7, Section 3, Theorem] to construct anorthonormal sequence (finite or infinite) which will be used to define the best compact approximant fromthe closed unit ball of K ( H ).Let e ∈ H be a unit vector such that k T ( e ) k = k T k and P be the projection onto { e } ⊥ . Havingchosen the orthonormal set { e , e , ..., e n } , let P n be the orthogonal projection onto { e , e , ..., e n } ⊥ = E n .Let E = H and P be the identity operator on H . Since T is not compact and I − P n is a finite rankprojection, k T P n k 6 = 0. If T P n attains its norm, then choose a unit vector e n +1 such that k T P n ( e n +1 ) k = k T P n k . If T P n does not attain its norm, then we stop the process of choosing unit vectors and proceedwith the finite set { e , . . . , e n } to obtain a best approximation as we shall see in Subcase 1 below. On theother hand, if T P n attains its norm at e n +1 for all n , then we proceed with the infinite set { e , e , . . . } aswe shall see in Subcase 2 below. Subcase 1:
Suppose that P m exists only for finitely many m . That is, there exists an n ≥ T P k − attains its norm at e k for k = 1 , , ..., n and T P n does not attain its norm.In this case, by a similar argument as in the proof of Case 3 of [7, Section 3, Theorem], we get that P k − ( e k ) = e k , e k ∈ E k − and h T ( e k ) , T P k ( x ) i = 0 for all x ∈ H and k = 1 , , ..., n . C. R. JAYANARAYANAN AND SREEJITH SIJU
In particular, for 1 ≤ i < j ≤ n , h T ( e i ) , T ( e j ) i = h T ( e i ) , T P i ( e j ) i = 0 and h T ( e k ) , T P n ( x ) i = h T ( e k ) , T P k ( P n ( x )) i = 0 for k = 1 , , ..., n. Now define an operator L : H → H as L = 1 k T k n X i =1 T e i ⊗ e i , where ( T e i ⊗ e i )( x ) = h x, e i i T e i .Since h Lx, Lx i = k T k P ni =1 |h x, e i i| k T e i k ≤ k x k for every x ∈ H , we have k L k ≤ x ∈ B H . Write x = u + v , where u ∈ span { e , e , ..., e n } and v ∈ { e , e , ..., e n } ⊥ . Then Lv = 0 and h ( T − L ) u, T v i = h ( T − L )( P ni =1 h u, e i i e i ) , T P n v i = DP ni =1 (1 − k T k ) h u, e i i T e i , T P n v E = (cid:18) − k T k (cid:19) P ni =1 h u, e i ih T e i , T P n v i = 0 . In addition, by Case 2 of [7, Section 3, Theorem], we have k T P n k = ∆( T ). Therefore, k ( T − L )( x ) k = h ( T − L )( u + v ) , ( T − L )( u + v ) i = h ( T − L ) u, ( T − L ) u i + h T v, T v i = * n X i =1 (cid:18) k T k − k T k (cid:19) h u, e i i T e i , n X i =1 (cid:18) k T k − k T k (cid:19) h u, e i i T e i + + h T P n v, T P n v i = (cid:18) k T k − k T k (cid:19) n X i =1 |h u, e i i| k T e i k + k T P n v k ≤ ( k T k − k u k + ∆( T ) k v k ≤ max { ( k T k − , ∆( T ) }k x k . Thus max {k T k − , ∆( T ) } ≤ d ( T, B K ( H ) ) ≤ k T − L k ≤ max {k T k − , ∆( T ) } . So d ( T, B K ( H ) ) = k T − L k .Hence L is a best approximation to T from B K ( H ) . Subcase 2:
Suppose P n exists for all n ∈ N . That is, T P n attains its norm at e n +1 for all n ∈ N .In this case, by a similar argument as in Case 3 of [7, Section 3, Theorem], we can see that theorthonormal sequence of vectors ( e n ) ∞ n =1 satisfies the following properties. P n ( e n +1 ) = e n +1 and hence k T P n k = k T e n +1 k for all n ∈ N , (2.1) h T ( e n ) , T P n ( x ) i = 0 for all n ∈ N and for any x ∈ H, (2.2) h T ( e i ) , T ( e j ) i = 0 for all i = j ,(2.3) lim n →∞ k T e n k = ∆( T ) and(2.4) ∆( T ) ≤ k T e n k for all n. (2.5) ALL PROXIMINALITY OF M -IDEALS OF COMPACT OPERATORS 5 Since T is not compact, ∆( T ) > k T e n k > n . Now define an operator L : H −→ H as L = 1 k T k K X n =1 T e n ⊗ e n + ∞ X n = K +1 ( k T e n k − ∆( T )) T e n k T e n k ⊗ e n , where K is chosen so that k T e n k − ∆( T ) ≤ n ≥ K . Since the sequence ( k T e n k − ∆( T )) convergesto 0 and ( T e n k T e n k ) forms an orthonormal sequence, by [11, Page 8, Theorem 1] and [11, Page 13, Corollary], L is a compact operator with k L k ≤ x ∈ B H . Write x = u + v , where u ∈ span { e , e , ..., } and v ∈ { e , e , .... } ⊥ . Define λ n = k T k− k T k h u, e n i if n ≤ K, ∆( T ) k T e n k h u, e n i if n > K. Then ( T − L ) u = ( T − L )( P ∞ n =1 h u, e n i e n ) = P ∞ n =1 λ n T e n . Since v ∈ { e , e , . . . } ⊥ , P n v = v for all n ∈ N .Now, using (2.2), we get h T v, ( T − L ) u i = * T v, ∞ X n =1 λ n T e n + = ∞ X n =1 h T v, λ n T e n i = ∞ X n =1 h T P n v, λ n T e n i = 0 . (2.6)Since k T v k = k T P n v k ≤ k T P n k k v k ≤ k T e n +1 k k v k for every n , by (2.4), we get k T v k ≤ ∆( T ) k v k .This together with (2.6) and using the fact that Lv = 0, we get k ( T − L ) x k = h ( T − L ) u, ( T − L ) u i + 2 Re h T v, ( T − L ) u i + h T v, T v i = * K X n =1 λ n T e n + ∞ X n = K +1 λ n T e n , K X n =1 λ n T e n + ∞ X n = K +1 λ n T e n + + k T v k ≤ * K X n =1 λ n T e n , K X n =1 λ n T e n + + * ∞ X n = K +1 λ n T e n , ∞ X n = K +1 λ n T e n + + ∆( T ) k v k ≤ ( k T k − K X n =1 |h u, e n i| + ∞ X n = K +1 ∆( T ) |h u, e n i| + ∆( T ) k v k ≤ max (cid:8) ( k T k − , ∆( T ) (cid:9) k x k . Thus d ( T, B K ( H ) ) = k T − L k = max {k T k − , ∆( T ) } . Hence L is a best approximation to T from B K ( H ) . Case 2: k T k ≤ T is not compact, by [7, Section 3, Theorem], we can choose an element L ∈ P K ( H ) ( T ) such that L is either of the form L = P ∞ n =1 ( k T ( e n ) k − ∆( T )) T e n k T e n k ⊗ e n for some orthonormal sequence ( e n ) with k T ( e n ) k > n or L = T P for some finite rank orthogonal projection P on H . Since k T k ≤ k L k ≤
1. Then d ( T, B K ( H ) ) ≤ k T − L k = d ( T, K ( H )) ≤ d ( T, B K ( H ) ). Thus d ( T, B K ( H ) ) = max {k T k − , ∆( T ) } = k T − L k . Thus L is a best approximation to T from B K ( H ) and hence K ( H ) is ball proximinal in B ( H ). (cid:3) C. R. JAYANARAYANAN AND SREEJITH SIJU
We now prove that the distance d ( T, B K ( H ) ) coincides with d ( T, K ( H )) when T is a scalar multipleof an extreme point of closed unit ball of B ( H ). Towards this, we describe P B X ( x ) when x is a scalarmultiple of an extreme point of the closed unit ball of a Banach space X . Theorem 2.2.
Let e be an extreme point of the closed unit ball of a Banach space X . Then P B X ( αe ) = { αe | α | } for all α ∈ C with | α | > .Proof. Assume first that α ∈ R and α >
1. Let e be an extreme point of B X . Then − e is an extreme pointof B X . So (1 − α ) e is an extreme point of B (0 , α − αe we get that e is an extremepoint of B ( αe, α − . Now suppose that there exists an f ∈ B X such that k αe − f k = α − d ( αe, B X ).Since e, f ∈ B ( αe, α − te + (1 − t ) f ∈ B ( αe, α −
1) for all t ∈ [0 , y = (2 − t ) e − (1 − t ) f ,where t ∈ (0 ,
1) is chosen so that α + t >
2. Then k αe − ((2 − t ) e − (1 − t ) f ) k ≤ | α − (2 − t ) | + | − t | ≤ | α + t − | + 1 − t ≤ α − . Hence y ∈ B ( αe, α − y + te +(1 − t ) f = (2 − t ) e − (1 − t ) f + te +(1 − t ) f = e = e . Since e is an extremepoint of B ( αe, α − y = te + (1 − t ) f . That is, (2 − t ) e − (1 − t ) f = te + (1 − t ) f. Hence e = f. For | α | >
1, if we let f = αe | α | , then f is again an extreme point of B X . Hence by the first part of theproof we get P B X ( αe ) = P B X ( | α | f ) = { f } = { αe | α | } . (cid:3) For a Hilbert space H , it is well known that the extreme points of the closed unit ball of B ( H ) areprecisely the isometries and co-isometries. Now the distance formula (1) in Theorem 2.1 together withTheorem 2.2 gives the following result. Corollary 2.3.
Let H be an infinite dimensional Hilbert space and V ∈ B ( H ) be an isometry or aco-isometry. Then for each a ∈ C , d ( aV, B K ( H ) ) = d ( aV, K ( H )) .Proof. If | a | ≤
1, then by Theorem 2.1, d ( aV, B K ( H ) ) = d ( aV, K ( H )). Now suppose that | a | > d ( aV, B K ( H ) ) = d ( aV, K ( H )). Then, by Theorem 2.1, there exists a K ∈ B K ( H ) such that d ( aV, B K ( H ) ) = k aV − K k = | a | − d ( aV, B B ( H ) ). Hence, by Theorem 2.2, K = aV | a | , which is a contradiction since V is not a compact operator. (cid:3)
3. Ball proximinality of M-ideals of compact operators and M-embedded spaces.In this section, we prove the ball proximinality of K ( X ) in B ( X ) when X is a Banach space suchthat K ( X ) is an M -ideal in B ( X ). In this direction, we begin with a lemma which is a modification of[13, Proposition 2.3].Note that if Y is an M -ideal in X , then every y ∗ ∈ Y ∗ has a unique norm preserving extension to afunctional x ∗ ∈ X ∗ . Thus we can consider Y ∗ as a subspace of X ∗ . So it make sense to define the weaktopology on X induced by Y ∗ . We will denote this by σ ( X, Y ∗ ). ALL PROXIMINALITY OF M -IDEALS OF COMPACT OPERATORS 7 Lemma 3.1.
Let J be an M -ideal in a Banach space X and x ∈ X . Then there exists a net ( y α ) in J such that ( y α ) converges to x k x k in the σ ( X, J ∗ ) -topology, and for each z ∈ X and ε > there exists anindex α such that (3.1) k z + β ( x − y α ) k ≤ ε + max {k z + k x k − k x k βx k , k z + J k + β k x + J k} for every α ≥ α and β ∈ [0 , . Consequently, lim sup α k z + β ( x − y α ) k ≤ max {k z + k x k − k x k βx k , k z + J k + β k x + J k} for every z ∈ X and β ∈ [0 , . Proof.
We follow the proof technique of [13, Proposition 2.3]. Let Q denote the M -projection from X ∗∗ onto J ⊥⊥ . Then k z + x − Qx k x k k = k Q ( z + k x k − k x k x ) + ( I − Q )( z + x ) k≤ max {k ( z + k x k − k x k x ) k , k z + J k + k x + J k} for every z ∈ X. Let A be the set of all triplets α = ( E, F, ε ) where E ⊂ X ∗∗ and F ⊂ X ∗ are finite dimensionalsubspaces and ε >
0. Then A can be partially ordered as: ( E , F , ε ) ≤ ( E , F , ε ) if E ⊆ E , F ⊆ F and ε ≤ ε . Then corresponding to each triplet α , by principle of local reflexivity ([4, Theorem3.2]), there exists an operator T α such that k T α k (1 + ε ), T α | E ∩ X = Id, T α (cid:0) E ∩ J ⊥⊥ (cid:1) ⊂ J and h T α x ∗∗ , x ∗ i = h x ∗∗ , x ∗ i for every x ∗∗ ∈ E and x ∗ ∈ F . Define y α = T α ( Qx k x k ). Then, by proceeding as inthe proof of [13, Proposition 2.3], we can see that y α → x k x k in the σ ( X, J ∗ )-topology. Assume now that ε > z ∈ X . Write δ = max {k z k + |k x k − | , k z + J k + k x + J k} . Let α = ( E , F , εδ ),where E and F are finite dimensional subspaces of X ∗∗ and X ∗ respectively such that z, x ∈ E .Now for any 1 ≥ β > α ≥ α , (cid:13)(cid:13)(cid:13)(cid:13) zβ + x − y α (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) T α ( zβ + x − Qx k x k ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:16) εδ (cid:17) (cid:13)(cid:13)(cid:13)(cid:13) zβ + x − Qx k x k (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:16) εδ (cid:17) max (cid:26)(cid:13)(cid:13)(cid:13)(cid:13) zβ + k x k − k x k x (cid:13)(cid:13)(cid:13)(cid:13) , β k z + J k + k x + J k (cid:27) ≤ max (cid:26)(cid:13)(cid:13)(cid:13)(cid:13) zβ + k x k − k x k x (cid:13)(cid:13)(cid:13)(cid:13) , β k z + J k + k x + J k (cid:27) + εβδ δ. Therefore, k z + β ( x − y α ) k = β k zβ + x − y α k ≤ ε + max (cid:26) k z + k x k − k x k βx k , k z + J k + β k x + J k (cid:27) for every α ≥ α and 0 < β ≤ (cid:3) Now by applying Lemma 3.1 to M -ideals of compact operators and by following the arguments usedin the proof of (i) = ⇒ (ii) of [13, Theorem 3.1], we obtain the inequality in the following lemma whichis basically a modification of the revised basic inequality studied in [13]. C. R. JAYANARAYANAN AND SREEJITH SIJU
Lemma 3.2.
Let X be a Banach space such that K ( X ) is an M -ideal in B ( X ) and let T ∈ B ( X ) . Thenthere exists a net ( L α ) in K ( X ) such that L ∗ α −→ T ∗ k T k in WOT, and for each S ∈ B ( X ) and ε > thereexists an α such that (3.2) k S + β ( T − L α ) k ≤ ε + max (cid:26)(cid:13)(cid:13)(cid:13)(cid:13) S + ( k T k − k T k βT (cid:13)(cid:13)(cid:13)(cid:13) , k S k e + β k T k e (cid:27) for every α ≥ α and < β ≤ . Consequently, lim sup α k S + β ( T − L α ) k ≤ max (cid:26)(cid:13)(cid:13)(cid:13)(cid:13) S + ( k T k − k T k βT (cid:13)(cid:13)(cid:13)(cid:13) , k S k e + β k T k e (cid:27) for all S ∈ B ( X ) and β ∈ [0 , . We now prove the main theorem of this section. We prove the ball proximinality of K ( X ) when it isan M -ideal in B ( X ). For an operator T with k T k >
1, we construct, using the basic inequality method, asequence ( λ i ) of positive numbers and a sequence ( T α ( i ) ) of compact operators such that K = P ∞ i =1 λ i T α ( i ) is a best approximation to T from the closed unit ball of K ( X ). Theorem 3.3.
Let X be a Banach space such that K ( X ) is an M -ideal in B ( X ) . Then K ( X ) is ballproximinal in B ( X ) . Moreover, d ( T, B K ( X ) ) = max {k T k − , d ( T, K ( X )) } for all T ∈ B ( X ) . Proof.
Let T ∈ B ( X ). Without loss of generality, we may assume that T is not a compact operator. For,if T is a compact operator then T k T k is a best approximation to T from B K ( X ) .Since K ( X ) is an M -ideal in B ( X ), by [6, Chapter VI, Proposition 4.10], there exists a net ( K α ) ofcompact operators with k K α k ≤ K ∗ α ) converges to T ∗ k T k in SOT. Let ( L α ) be the net devised fromLemma 3.2. Then ( K ∗ α − L ∗ α ) converges to 0 in WOT. Hence ( K α − L α ) converges to 0 in the weak topology σ ( K ( X ) , K ( X ) ∗ ). Thus k T α − S α k → T α ∈ conv { K β | β > α } and S α ∈ conv { L β | β > α } , whereconv denotes the convex hull. Since ( S α ) satisfies inequality (3.2), the net ( T α ) of compact operators with k T α k ≤ k T k >
1. Recall that d ( T, K ( X )) = k T k e . Let 0 < a < a k T k − >
0. Pick a strictly decreasing sequence ( ε i ) of positive numbers such that P ∞ i =1 ε i converges and a ≤ max {k T k− , k T k e }− P ∞ i =1 ε i max {k T k− , k T k e } .Now by using a technique similar to the one used in [1, Theorem 1], we inductively construct a sequence( λ i ) of positive numbers and a sequence ( T α ( i ) ) of compact operators from the net ( T α ) such that ∞ X i =1 λ i ≤ , (3.3) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = max {k T k − , k T k e } − ε n for n = 1 , , . . . . (3.4) k λ ( T − T α (1) ) k ≤ ε + max { λ ( k T k − , λ k T k e } , (3.5a) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n +1 X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε n +1 + max ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( T − T α ( i ) ) + λ n +1 ( k T k − T k T k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , ( n +1 X i =1 λ i ) k T k e ) (3.5b) ALL PROXIMINALITY OF M -IDEALS OF COMPACT OPERATORS 9 for n = 1 , , . . . .We start the construction by taking S = 0 in Lemma 3.2. Then there exists an α (1) such that (cid:13)(cid:13) β ( T − T α (1) ) (cid:13)(cid:13) ≤ ε + max (cid:26)(cid:13)(cid:13)(cid:13)(cid:13) ( k T k − k T k βT (cid:13)(cid:13)(cid:13)(cid:13) , β k T k e (cid:27) for all β ∈ [0 , . Now let λ > k λ ( T − T α (1) ) k = max {k T k − , k T k e } − ε .Suppose we have chosen scalars λ , λ , ..., λ n and indices α (1) , α (2) , ..., α ( n ) such that α (1) < α (2) <... < α ( n ) and(3.6) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = max {k T k − , k T k e } − ε n . Let S = P ni =1 λ i ( T − T α ( i ) ). Then, by Lemma 3.2, there exists an α ( n + 1) > α ( n ) such that(3.7) (cid:13)(cid:13) S + β ( T − T α ( n +1) ) (cid:13)(cid:13) ≤ ε n +1 + max (cid:26)(cid:13)(cid:13)(cid:13)(cid:13) S + β ( k T k − T k T k (cid:13)(cid:13)(cid:13)(cid:13) , k S k e + β k T k e (cid:27) for all β ∈ [0 , . Now, to obtain λ n +1 , we consider the quantity:(3.8) (cid:13)(cid:13) S + β ( T − T α ( n +1) ) (cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( T − T α ( i ) ) + β ( T − T α ( n +1) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . For β = 0, by the induction hypothesis (3.6), the quantity (3.8) becomes max {k T k − , k T k e } − ε n . As β −→ ∞ , (3.8) becomes larger than max {k T k − , k T k e } − ε n +1 . Hence there exists a λ n +1 such that(3.9) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n +1 X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = max {k T k − , k T k e } − ε n +1 . Then, from (3.9), we obtainmax {k T k − , k T k e }− ε n +1 = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n +1 X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( n +1 X i =1 λ i ) T − n +1 X i =1 λ i T α ( i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ ( n +1 X i =1 λ i ) max {k T k − , k T k e } . Thus P n +1 i =1 λ i ≤ β = λ n +1 in (3.7). Then we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n +1 X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( T − T α ( i ) ) + λ n +1 ( T − T α ( n +1) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε n +1 + max ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( T − T α ( i ) ) + λ n +1 ( k T k − T k T k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e + λ n +1 k T k e ) ≤ ε n +1 + max ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( T − T α ( i ) ) + λ n +1 ( k T k − T k T k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , ( n X i =1 λ i ) k T k e + λ n +1 k T k e ) ≤ ε n +1 + max ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( T − T α ( i ) ) + λ n +1 ( k T k − T k T k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , ( n +1 X i =1 λ i ) k T k e ) . Hence, by induction, we obtain a sequence ( λ i ) of positive numbers and a sequence ( T α ( i ) ) of compactoperators satisfying (3.3), (3.4), (3.5a), (3.5b).Since P ∞ i =1 λ i ≤
1, it is easy to see that the series P ∞ i =1 λ i T α ( i ) converges. Let K = P ∞ i =1 λ i T α ( i ) . Then K is a compact operator and k K k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = max {k T k − , k T k e } . If P ∞ i =1 λ i = 1, then from (3.10) and using the fact that max {k T k − , k T k e } ≤ d ( T, B K ( X ) ), we get d ( T, B K ( X ) ) ≤ k T − K k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = max {k T k − , k T k e } ≤ d ( T, B K ( X ) ) . Hence K is a best approximation to T from the closed unit ball of K ( X ).So it is enough to prove that P ∞ i =1 λ i ≥
1. To obtain this, we split the proof into two cases dependingon where the maximum occurs in the inequality (3.5b).
Case 1:
Assume that the maximum in the inequality (3.5b) is attained at the second term forinfinitely many n . That is, there exists an increasing sequence ( n k ) of positive integers such that (cid:13)(cid:13)(cid:13)P n k +1 i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13) ≤ ε n k +1 + ( P n k +1 i =1 λ i ) k T k e ≤ ε n k +1 + ( P ∞ i =1 λ i ) k T k e . Thenmax {k T k − , k T k e } = lim k →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n k X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ lim k →∞ ε n k + ( ∞ X i =1 λ i ) k T k e , Hence max {k T k − , k T k e } ≤ ( P ∞ i =1 λ i ) k T k e and therefore ( P ∞ i =1 λ i ) ≥ . Case 2:
Assume that the maximum in the inequality (3.5b) is attained at the second term only forfinitely many n .We prove the case when the maximum in the inequality (3.5b) is attained at the second term for atleast one n ∈ N or maximum in the inequality (3.5a) is attained at the second term. A similar proofholds even if the maximum in both inequalities (3.5a) and (3.5b) is not attained at the second term forany n ∈ N . Hence we assume that there exists a positive integer N such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε N +( N X i =1 λ i ) k T k e , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε m + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m − X i =1 λ i ( T − T α ( i ) ) + λ m ( k T k − T k T k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ALL PROXIMINALITY OF M -IDEALS OF COMPACT OPERATORS 11 for all m > N. Now for any m > N , ε m + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m − X i =1 λ i ( T − T α ( i ) ) + λ m ( k T k − T k T k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε m + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m − X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13) λ m ( k T k − T k T k (cid:13)(cid:13)(cid:13)(cid:13) ≤ ε m + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m − X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + λ m ( k T k − ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m − k = N X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + m X i = N +1 λ i ( k T k −
1) + m X i = N +1 ε i . Now using (3.10) and letting m → ∞ in the above inequality, we can see thatmax {k T k − , k T k e } = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = lim m →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X i =1 λ i ( T − T α ( i ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + ∞ X i = N +1 λ i ( k T k −
1) + ∞ X i = N +1 ε i ≤ ε N + ( N X i =1 λ i ) k T k e + ∞ X i = N +1 λ i ( k T k −
1) + ∞ X i = N +1 ε i ≤ ∞ X i =1 λ i max {k T k − , k T k e } + ∞ X i =1 ε i . (3.11)Thus a ≤ max {k T k− , k T k e }− P ∞ i =1 ε i max {k T k− , k T k e } ≤ P ∞ i =1 λ i ≤ P ∞ i =1 λ i ≥
1. Let b = P ∞ i =1 λ i .Suppose b <
1. Then, by (3.10), k bT − K k = max {k T k − , k T k e } . Take t = 1 − b k T k− k T k . Then0 < t < k bT − ( tbT + (1 − t ) K ) k = (1 − t ) max {k T k − , k T k e } ≤ (1 − t ) k T k ≤ b k T k − . Now take λ = tb k T k +1 . Since 0 < λ < k bT − T k T k k = b k T k −
1, we have (cid:13)(cid:13)(cid:13)(cid:13) bT − (cid:18) λ ( tbT + (1 − t ) K ) + (1 − λ ) T k T k (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ b k T k − . Since λ ( tbT + (1 − t ) K ) + (1 − λ ) T k T k = (cid:16) λtb − (1 − λ ) k T k (cid:17) T + λ (1 − t ) K = λ (1 − t ) K , it follows that k bT − λ (1 − t ) K k ≤ b k T k −
1. Thus λ (1 − t ) K is a best approximation to bT from the closed unit ball of B ( X ) and k λ (1 − t ) K k < b <
1, which is a contradiction as a best approximation to bT from the closedunit ball of B ( X ) always has norm 1. Hence b = P ∞ i =1 λ i = 1.Now let T ∈ B ( X ) be such that k T k ≤
1. Then, by [6, Chapter VI, Proposition 4.10], there exists anet ( K α ) of compact operators with k K α k ≤ K ∗ α ) converges to T ∗ in SOT. Then, by [13, Corollary3.2], there exists a compact operator K ∈ conv { K α } such that k T − K k = d ( T, K ( X )), where conv { K α } denotes the norm closure of conv { K α } . Since k K α k ≤
1, we get k K k ≤ d ( T, B K ( X ) ) = d ( T, K ( X )) = k T − K k . Hence K ( X ) is ball proximinal in B ( X ). (cid:3) We now state a remark which follows from [1, Theorem 1].
Remark . Let H be a separable Hilbert space. Then for each positive (self-adjoint) operator T on H , there exists a compact positive (self-adjoint) operator K such that d ( T, K ( H )) = k T − K k . For,let { e , e , ... } be an orthonormal basis for H and P n be the orthogonal projection onto { e , e , ..., e n } .Put T n = P n T P n for n ≥
1. Then ( T n ) is a sequence of compact positive (self-adjoint) operator suchthat T n → T in SOT and T ∗ n → T ∗ in SOT. Now, by [1, Theorem 1], there exists a sequence ( a n ) ofnon-negative real numbers such that K = P ∞ n =1 a n T n is a compact positive (self-adjoint) operator thatsatisfies d ( T, K ( H )) = k T − K k .The following result shows that positive (self-adjoint) operators on a Hilbert space H has a positive(self-adjoint) compact approximant from the closed unit ball of K ( H ). Corollary 3.5.
Let H be a separable Hilbert space. Then for each positive (self-adjoint) operator T on H with k T k ≥ , there exists a positive (self-adjoint) compact operator K on H such that k K k ≤ and d ( T, B K ( H ) ) = k T − K k . Proof.
Let { e , e , ... } be an orthonormal basis for H and P n be the orthogonal projection onto { e , e , ..., e n } for n ≥
1. Put K n = P n T k T k P n for n ≥
1. Then ( K n ) is a sequence of positive (self-adjoint) compactoperators such that k K n k ≤ K n → T k T k in SOT and K ∗ n → T ∗ k T k in SOT. Now, by proceeding as in theproof of Theorem 3.3, we get a sequence ( λ n ) of non-negative real numbers such that K = P ∞ n =1 λ n T n is a compact positive (self-adjoint) operator that satisfies k K k ≤ k T − K k = d ( T, B K ( H ) ) , where T n ∈ conv { K m : m ≥ } . (cid:3) We observe from the proof of Theorem 3.3 that if Y is an M -ideal in X and if there is a net ( y α ) suchthat k y α k ≤ y α ) satisfies the inequality (3.1) of Lemma 3.1, then Y is ball proximinal in X andfor x ∈ X , d ( x, B Y ) = max {k x k − , d ( x, Y ) } . We now use this fact to prove that M -embedded spacesare ball proximinal in its bidual. Theorem 3.6.
Let X be an M -embedded space. Then X is ball proximinal in X ∗∗ and d ( x ∗∗ , B X ) =max {k x ∗∗ k − , d ( x ∗∗ , X ) } for x ∗∗ ∈ X ∗∗ .Proof. Let x ∗∗ ∈ X ∗∗ be such that k x ∗∗ k > y α ) be the net in X devised from Lemma 3.1. Since B X is weak ∗ -dense in B X ∗∗ , there exists a net ( x α ) in X such that k x α k ≤ x α ) converges to x ∗∗ k x ∗∗ k in theweak*-topology. Then the net ( x α − y α ) converges to 0 in the weak*- topology of X ∗∗ and hence ( x α − y α )converges to 0 in the weak topology of X . Therefore k z α − u α k → z α ∈ conv { x β : β ≥ α } and u α ∈ conv { y β : β ≥ α } . Clearly k z α k ≤
1. Since ( u α ) satisfies inequality (3.1), it follows that the net ( z α )satisfies inequality (3.1). Now by proceeding as in the proof of of Theorem 3.3, we can construct a sequence( λ n ) of non-negative real numbers and a sequence ( z α n ) from the net ( z α ) such that z = P ∞ n =1 λ n z α n isa best approximation to x ∗∗ from B X and d ( x ∗∗ , B X ) = max {k x ∗∗ k − , d ( x ∗∗ , X ) } = k x ∗∗ − z k .If k x ∗∗ k ≤
1, then there exists a net ( x α ) in B X such that ( x α ) converges to x ∗∗ in the weak* topology.Since the weak* topology on X ∗∗ coincide with σ ( X ∗∗ , X ∗ )-topology, by the remark following [13, Corol-lary 3.2], there exists an element y ∈ conv { x α } such that d ( x ∗∗ , X ) = k x − y k . Since k x α k ≤ α ,we have y ∈ B X and hence d ( x ∗∗ , B X ) = d ( x ∗∗ , X ) = k x − y k . Therefore X is ball proximinal in X ∗∗ . (cid:3) ALL PROXIMINALITY OF M -IDEALS OF COMPACT OPERATORS 13 Remark . If X is a reflexive space such that K ( X ) is an M -ideal in B ( X ), then, by [6, Chapter VI,Proposition 4.11], B ( X ) is isometric to the bidual of K ( X ) and hence, by Theorem 3.3, K ( X ) is ballproximinal in its bidual.We now give an example of a Banach space X such that K ( X ) is ball proximinal in B ( X ), but K ( X )is not an M -ideal in B ( X ). It is well known that K ( ℓ ) is not an M -ideal in B ( ℓ ) (see [6]). Example 3.8. K ( ℓ ) is ball proximinal in B ( ℓ ) and d ( T, B K ( ℓ ) ) = max {k T k − , d ( T, K ( ℓ )) } for T ∈B ( ℓ ). Proof.
We know from [12, Page 220] that every operator T on ℓ has a matrix representation [ t ij ] withrespect to the canonical basis { e i } ∞ .Let T = [ t ij ] ∈ B ( ℓ ). Then, by [9, Theorem 1.3], d ( T, K ( ℓ )) = R , where R = lim n →∞ sup j P ∞ i = n | t ij | .Now let d = max {k T k− , R } . Clearly, d ≤ d ( T, B K ( ℓ ) ). We follow a similar construction as in [9, Theorem1.3] to obtain a compact operator K = [ k ij ] such that k T − K k = d ( T, B K ( ℓ ) ) = d and k K k ≤ j ∈ N , the j th column of K = [ k ij ] is defined as follows:If P ∞ i =1 | t ij | d , set k ij = 0 for i = 1 , . . . , ∞ .If P ∞ i =1 | t ij | > d, let n be the largest index such that P ∞ i = n | t ij | > d . Then there exists a real number a ∈ [0 ,
1] such that a | t nj | + P ∞ i = n +1 | t ij | = d . Now set k ij as: k ij = t ij if 1 ≤ i < n, (1 − a ) t ij if i = n, i > n. Now to prove k K k ≤
1, we assume n > n = 1). Then for j with P ∞ i =1 | t ij | > d ,since P n − i =1 | t ij | + a | t nj | + (1 − a ) | t nj | + P ∞ i = n +1 | t ij | = P ∞ i =1 | t ij | ≤ k T k and d ≥ k T k −
1, we get ∞ X i =1 | k ij | = n − X i =1 | t ij | + (1 − a ) | t nj | ≤ k T k − d ≤ . Therefore k K k ≤ k T − K k = sup j P ∞ i =1 | t ij − k ij | = d . Now a similar argument as in theproof of [9, Theorem 1.3] gives the compactness of the operator K . Thus d ( T, B K ( ℓ ) ) = k T − K k =max {k T k − , d ( T, K ( ℓ )) } and hence K ( ℓ ) is ball proximinal in B ( ℓ ). (cid:3) AcknowledgementsThe research of the first named author is supported by SERB MATRICS grant(No. MTR/2017/000926)and the research of the second named author is supported by UGC Junior research fellowship (No.20/12/2015(ii)EU-V).
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