On the existence of non-norm-attaining operators
OON THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS
SHELDON DANTAS, MINGU JUNG, AND GONZALO MART´INEZ-CERVANTES
Abstract.
In this paper we provide necessary and sufficient conditions for the existenceof non-norm-attaining operators in L ( E, F ). By using a theorem due to Pfitzner on Jamesboundaries, we show that if there exists a relatively compact set K of L ( E, F ) (in theweak operator topology) such that 0 is an element of its closure (in the weak operatortopology) but it is not in its norm closed convex hull, then we can guarantee the existenceof an operator which does not attain its norm. This allows us to provide the followinggeneralization of results due to Holub and Mujica. If E is a reflexive space, F is anarbitrary Banach space, and the pair ( E, F ) has the bounded compact approximationproperty, then the following are equivalent:(i) K ( E, F ) = L ( E, F );(ii) Every operator from E into F attains its norm;(iii) ( L ( E, F ) , τ c ) ∗ = ( L ( E, F ) , (cid:107)·(cid:107) ) ∗ ;where τ c denotes the topology of compact convergence. We conclude the paper presentinga characterization of the Schur property in terms of norm-attaining operators. Introduction
The famous James theorem states that a Banach space E is reflexive if and only if everybounded linear functional on E attains its norm. By using this characterization, one cancheck that if every bounded linear operator from a Banach space E into a Banach space F is norm-attaining, then E must be reflexive, whereas the range space F is not forced tobe reflexive in general. Indeed, every bounded linear operator from a reflexive space intoa Banach space which satisfies the Schur property is compact and any compact operatorfrom a reflexive space into an arbitrary Banach space is norm-attaining. Therefore, itseems natural to wonder whether it is possible to guarantee the existence of a non-norm-attaining operator from the existence of a non-compact operator. This brings us back tothe 70’s when J.R. Holub proved that this is, in fact, true under approximation propertyassumptions (see [15, Theorem 2]). Almost thirty years later, J. Mujica improved Holub’sresult by using the compact approximation property (see [24, Theorem 2.1]), which is aweaker assumption than the approximation property. However, both results require thereflexivity on both domain and range spaces, so the following question arises naturally. Mathematics Subject Classification.
Key words and phrases.
James theorem; norm-attatining operators; compact approximation property.S. Dantas was supported by the project OPVVV CAAS CZ.02.1.01/0.0/0.0/16 019/0000778 and bythe Estonian Research Council grant PRG877. The second author was supported by NRF (NRF-2019R1A2C1003857). The third author was partially supported by
Fundaci´on S´eneca [20797/PI/18],
Agencia Estatal de Investigaci´on [MTM2017-86182-P, cofunded by ERDF, EU] and by the EuropeanSocial Fund (ESF) and the Youth European Initiative (YEI) under
Fundaci´on S´eneca [21319/PDGI/19]. a r X i v : . [ m a t h . F A ] F e b DANTAS, JUNG, AND MART´INEZ-CERVANTES
Given a reflexive space E and an arbitrary Banach space F , under which assumptionswe may guarantee the existence of non-norm-attaining operators in L ( E, F ) ? Coming back to Holub and Mujica’s results, we would like to highlight what theyproved in the direction of the above question. For a background on necessary definitionsand notations, we refer the reader to Section 2. In what follows, τ c denotes the topologyof compact convergence and (cid:107) · (cid:107) denotes the norm topology in L ( E, F ). Theorem ([15, Theorem 2] and [24, Theorem 2.1]) . Let E and F be both reflexive spaces.(a) If L ( E, F ) is non-reflexive, there is a non-norm-attaining operator S ∈ L ( E, F ).(b) If E or F has the (compact) approximation property, then the following statementsare equivalent.(i) There exists a non-norm-attaining operator S ∈ L ( E, F );(ii) L ( E, F ) (cid:54) = K ( E, F );(iii) L ( E, F ) is non-reflexive;(iv) ( L ( E, F ) , (cid:107) · (cid:107) ) ∗ (cid:54) = ( L ( E, F ) , τ c ) ∗ .The proof of the above result relies on the fact that if F is a reflexive space, then L ( E, F )is the dual space of the projective tensor product E (cid:98) ⊗ π F ∗ . However, if the range space F is non-reflexive, then L ( E, F ) is always non-reflexive (see, for instance, [29]).As a way of extending the above results to the case of non-reflexive range spaces, weborrow some of the techniques used by R.C. James (see [16, 17]). As a matter of fact, oneof his results [17, Theorem 1] implies that a separable Banach space E is non-reflexive ifand only if given 0 < θ <
1, there exists a sequence ( x ∗ n ) in B E ∗ such that x ∗ n w ∗ −→ , co { x ∗ n : n ∈ N } ) > θ , which in turn is equivalent to the existence of a relativelyweak* compact set K ⊆ B E ∗ such that 0 ∈ K w ∗ and 0 (cid:54)∈ co (cid:107)·(cid:107) ( K ). This motivates us todefine the following property. Definition 1.1.
We say that a pair (
E, F ) of Banach spaces has the
James property if there exists a relatively WOT-compact set K ⊆ L ( E, F ) such that 0 ∈ K W OT and0 (cid:54)∈ co (cid:107)·(cid:107) ( K ).We will prove that the James property is a sufficient condition to guarantee the existenceof an operator which does not attain its norm, which is our first aim in the present paper. Theorem A.
Let E and F be Banach spaces. If the pair ( E, F ) has the James property,then there exists a non-norm-attaining operator in L ( E, F ).Next, we prove that ( L ( E, F ) , (cid:107) · (cid:107) ) ∗ does not coincide with ( L ( E, F ) , τ c ) ∗ whenever apair ( E, F ) satisfies the James property (see Proposition 3.1). From this, we can see thatwhenever the pair (
E, F ) has the James property, then the Banach space L ( E, F ) cannotbe reflexive due to [24, Lemma 2.3].We observe, for a reflexive space E and an arbitrary Banach space F , that (1) theunit ball of K ( E, F ) is closed in the strong operator topology if and only if it is sequen-tially closed in this topology (see Lemma 3.4) and (2) K ( E, F ) = L ( E, F ) implies that( L ( E, F ) , (cid:107) · (cid:107) ) ∗ = ( L ( E, F ) , τ c ) ∗ by using the result [11, Theorem 1] due to M. Feder and N THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS 3
P. Saphar. Besides that, we consider the concept of the (bounded) compact approxima-tion property for a pair of Banach spaces in the way as it is done in [3] and prove that K ( E, F ) = L ( E, F ) when either (3) the norm-closed unit ball of K ( E, F ) is closed in thestrong operator topology or (4) ( L ( E, F ) , (cid:107) · (cid:107) ) ∗ = ( L ( E, F ) , τ c ) ∗ under the just mentionedapproximation property assumption (see Lemma 3.7). Combining (1)-(4) together withTheorem A, we get a generalization of Holub and Mujica’s results, where we no longerneed reflexivity on the target space F . Theorem B.
Let E be a reflexive space and F be an arbitrary Banach space. Considerthe following conditions.(a) K ( E, F ) = L ( E, F ).(b) Every operator from E into F attains its norm.(c) The unit ball B K ( E,F ) is closed in the strong operator topology.(d) ( L ( E, F ) , τ c ) ∗ = ( L ( E, F ) , (cid:107)·(cid:107) ) ∗ .Then, we always have (a) = ⇒ (b) = ⇒ (c) and (a) = ⇒ (d) = ⇒ (c). Additionally, if the pair( E, F ) has the bounded compact approximation property, then (c) = ⇒ (a) and thereforeall the statements are equivalent.The following diagram summarizes most of the results included in this article. In whatfollows, E is supposed to be any reflexive space and F is any arbitrary Banach space.Moreover, BCAP stands for the bounded compact approximation property for the pair( E, F ) (see Definition 2.1 in Section 2 below). B K ( E,F ) SOT (cid:54) = B K ( E,F ) ( E,F ) has the James property L ( E,F ) (cid:54) =NA( E,F ) L ( E,F ) (cid:54) = K ( E,F ) ( L ( E,F ) , (cid:107)·(cid:107) ) ∗ (cid:54) =( L ( E,F ) ,τ c ) ∗ L ( E,F ) is non-reflexiveIf (
E,F ) has the BCAP If F is reflexive Finally, as an application of Theorem B, we connect the Schur property with the casewhere every operator attains its norm, and obtain the following characterization (seeTheorem 3.10).
Theorem C.
Let F be a Banach space. The following statements are equivalent.(a) F has the Schur property.(b) K ( E, F ) = L ( E, F ) for every reflexive space E .(c) NA( E, F ) = L ( E, F ) for every reflexive space E .(d) K ( G, F ) = L ( G, F ) for every reflexive space G with basis.(e) NA( G, F ) = L ( G, F ) for every reflexive space G with basis. DANTAS, JUNG, AND MART´INEZ-CERVANTES Preliminaries
Throughout the paper, E and F will be Banach spaces over a field K , which canbe either the real or complex numbers. We denote by B E and S E the closed unit balland the unit sphere of the Banach space E , respectively. For a subset K of E , co( K )(resp., co( K )) denotes the convex hull (resp., closed convex hull) of K . The space of allbounded linear operators from E into F is denoted by L ( E, F ). The symbol K ( E, F )(resp., W ( E, F )) stands for the space of all compact operators (resp., weakly compactoperators) from E into F , whereas the symbol F ( E, F ) is used to denote the space ofall finite-rank operators. Recall that T ∈ L ( E, F ) is completely continuous if T sendsweakly null sequences in E to norm null sequences in F . We denote by V ( E, F ) the spaceof all completely continuous operators from E into F . Let us denote by W ∞ ( E, F ) thespace of all weakly ∞ -compact operators from E into F , which are introduced in [30].A subset C of a Banach space E is called relatively weakly ∞ -compact if there exists aweakly null sequence ( x n ) in E such that C ⊂ { (cid:80) ∞ n =1 a n x n : ( a n ) ∈ B (cid:96) } and an operator T ∈ L ( E, F ) is said to be weakly ∞ -compact if T ( B E ) is a relatively weakly ∞ -compactsubset of F . Finally, let us recall that an operator T ∈ L ( E, F ) attains its norm or it isnorm-attaining if there exists x ∈ B E such that (cid:107) T ( x ) (cid:107) = (cid:107) T (cid:107) . By NA( E, F ) we meanthe set of all norm-attaining operators from E into F . If E = F , then we simply writeNA( E ) instead of NA( E, E ) and we do the same with the above classes of operators.We will be using different topologies in L ( E, F ). We denote by τ c the topology ofcompact convergence, i.e. the topology of uniform convergence on compacts subsets of E .The weak operator topology (WOT, for short) is defined by the basic neighborhoods N ( T ; A, B, ε ) := (cid:8) S ∈ L ( E, F ) : | y ∗ ( T − S )( x ) | < ε, for every y ∗ ∈ B, x ∈ A (cid:9) , where A and B are arbitrary finite sets in E and F ∗ , respectively, and ε >
0. Thus, inthe WOT, a net ( T α ) converges to T if and only if ( y ∗ ( T α ( x ))) converges to y ∗ ( T ( x )) forevery x ∈ E and y ∗ ∈ F ∗ . Analogously, the strong operator topology (SOT, for short) isdefined by the basic neighborhoods N ( T ; A, ε ) := (cid:8) S ∈ L ( E, F ) : (cid:107) ( T − S )( x ) (cid:107) < ε, for every x ∈ A (cid:9) , where A is an arbitrary finite set in E and ε >
0. Thus, a net ( T α ) converges in theSOT to T if and only if ( T α ( x )) converges in norm to T ( x ) for every x ∈ E . We will dealwith SOT and WOT closures of bounded sets in L ( E, F ). It is worth mentioning that,for a bounded convex set in L ( E, F ), the WOT closure and the SOT closure coincide[9, Corollary VI.1.5]. Thus, the SOT and the WOT in some results in this paper canbe interchanged. For a more detailed exposition on topologies in L ( E, F ), we refer thereader to [7, 9].Let us present now the necessary definitions on approximation properties we will need.A Banach space E is said to have the approximation property (AP, for short) if theidentity operator Id E in L ( E ) belongs to F ( E ) τ c . Given λ (cid:62) E is said to have the λ -bounded approximation property ( λ -BAP, for short) when Id E belongs to λB F ( E ) τ c . ABanach space is said to have the bounded approximation property (BAP, for short) if ithas the λ -BAP for some λ (cid:62)
1. Especially, when λ = 1, we say that E has the metricapproximation property (MAP, for short). Also, recall that E is said to have the compact N THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS 5 approximation property (CAP, for short) if the identity operator Id E in L ( E ) belongs to K ( E ) τ c . The λ -bounded compact approximation property ( λ -BCAP, for short), boundedcompact approximation property (BCAP, for short) and metric compact approximationproperty (MCAP, for short) for a Banach space E can be defined in an analogous way.It is known that a reflexive space has the AP if and only if it has the MAP (see [13]).Analogously, every reflexive space with the CAP also has the MCAP (see [5, Proposition1 and Remark 1]). We refer the reader to [21, 22] and [4] for background.On the other hand, E. Bonde introduced in [3] the AP and λ -BAP for pairs of Banachspaces, that is, a pair ( E, F ) of Banach spaces is said to have the AP (resp., λ -BAP) ifany operator T ∈ L ( E, F ) belongs to F ( E, F ) τ c (resp., λB F ( E,F ) τ c for some λ (cid:62) E or F has the AP (resp., BAP), then the pair ( E, F ) has the AP (resp.,BAP). It is observed in [3, Section 4] that there are pairs of Banach spaces (
E, F ) withthe BAP such that E and F do not have the BAP. Similarly, we have the following. Definition 2.1.
The pair (
E, F ) of Banach spaces is said to have the compact approxima-tion property (CAP, for short) (resp., bounded compact approximation property (BCAP,for short)) if any operator T ∈ L ( E, F ) belongs to K ( E, F ) τ c (resp., λB K ( E,F ) τ c for some λ (cid:62) λ = 1, we say that the pair ( E, F ) has the metric compactapproximation property (MCAP, for short).As a matter of fact, [3, Example 4.2] shows that there are Banach spaces E and F suchthat ( E, F ) has the BCAP while E and F do not have the CAP. Thus, assuming that apair ( E, F ) of Banach spaces has CAP is more general than E or F has the CAP. We willbe using this fact without any explicit reference throughout the paper.3. The Results
In this section, we shall prove Theorems A, B, C, and their consequences. We start byproving Theorem A. To do so, let us recall that a set B ⊂ B E ∗ is called a James boundaryof a Banach space E if for every x ∈ S E , there exists f ∈ B such that f ( x ) = 1. For asubset G of E ∗ , we shall denote by w ( E, G ) the weak topology of X induced by G . Proof of Theorem A.
Let us assume by contradiction that every operator from E into F attains its norm. Then, the family B := (cid:110) x ⊗ y ∗ : x ∈ S E , y ∗ ∈ S F ∗ (cid:111) is a James boundary of L ( E, F ). Indeed, for an arbitrary operator T ∈ L ( E, F ) =NA(
E, F ), take x ∈ S E to be such that (cid:107) T ( x ) (cid:107) = (cid:107) T (cid:107) and then y ∗ ∈ S F ∗ to be such that | y ∗ ( T ( x )) | = (cid:107) T ( x ) (cid:107) = (cid:107) T (cid:107) . Now, since ( E, F ) has the James property, there exists arelatively WOT-compact set K ⊂ L ( E, F ) such that 0 ∈ K W OT and 0 (cid:54)∈ co (cid:107)·(cid:107) ( K ). By theUniform Boundedness principle, the set K W OT is norm-bounded. By hypothesis, K W OT is WOT-compact or, equivalently, w ( L ( E, F ) , B )-compact. By a theorem of Pfitzner (see[26] or [10, Theorem 3.121]), we have that K W OT is weakly compact. Therefore, 0 ∈ K W OT = K w , which in particular gives that 0 ∈ co w ( K ) = co (cid:107)·(cid:107) ( K ). This contradictionyields a non-norm-attaining operator T ∈ L ( E, F ) as desired. (cid:3)
DANTAS, JUNG, AND MART´INEZ-CERVANTES
Let us observe that if a pair (
E, F ) of Banach spaces has the James property, thenthe dual of L ( E, F ) endowed with the norm topology does not coincide with the dual of L ( E, F ) endowed with the topology τ c of compact convergence. As a matter of fact, if K is a subset of E given as in Definition 1.1, then there exists ϕ ∈ ( L ( E, F ) , (cid:107) · (cid:107) ) ∗ suchthat 0 = Re ϕ (0) > sup { Re ϕ ( T ) : T ∈ co( K ) } thanks to the Hahn-Banach separationtheorem. This implies that ϕ cannot be in ( L ( E, F ) , τ c ) ∗ since 0 ∈ co W OT ( K ) = co τ c ( K ).Moreover, using [24, Lemma 2.3], we see that if ( L ( E, F ) , (cid:107) · (cid:107) ) ∗ (cid:54) = ( L ( E, F ) , τ c ) ∗ , thenthe space L ( E, F ) cannot be reflexive. Summarizing, we obtain the following result.
Proposition 3.1.
Let E and F be Banach spaces. If the pair ( E, F ) has the Jamesproperty, then (i) ( L ( E, F ) , (cid:107) · (cid:107) ) ∗ (cid:54) = ( L ( E, F ) , τ c ) ∗ . (ii) L ( E, F ) is non-reflexive. One easy consequence of Theorem A is that if E is reflexive and a pair ( E, F ) has theJames property, then K ( E, F ) cannot be equal to the whole space L ( E, F ). As a matterof fact, the following result gives us a rather general observation.
Proposition 3.2.
Let E be a reflexive space and F be an arbitrary Banach space. If K ( E, F ) = L ( E, F ) , then ( L ( E, F ) , (cid:107) · (cid:107) ) ∗ = ( L ( E, F ) , τ c ) ∗ .Proof. Let D : E (cid:98) ⊗ π F ∗ −→ ( L ( E, F ) , τ c ) ∗ be defined by D ( z )( T ) := (cid:80) ∞ n =1 y ∗ n ( T ( x n )) forevery z ∈ E (cid:98) ⊗ π F ∗ with z = (cid:80) ∞ n =1 x n ⊗ y ∗ n and T ∈ L ( E, F ). It is well known that D is asurjective map (see, for example, [7, 5.5, pg. 62]). Therefore, we have that ( L ( E, F ) , τ c ) ∗ =( E (cid:98) ⊗ π F ∗ ) / ker D. On the other hand, from the result [11, Theorem 1], we have thatthe map V : E (cid:98) ⊗ π F ∗ −→ ( K ( E, F ) , (cid:107) · (cid:107) ) ∗ defined by V ( z )( T ) := (cid:80) ∞ n =1 y ∗ n ( T ( x n )) for z = (cid:80) ∞ n =1 x n ⊗ y ∗ n and T ∈ K ( E, F ), satisfies the following: for every ϕ ∈ ( K ( E, F ) , (cid:107) · (cid:107) ) ∗ ,there exists v ∈ E (cid:98) ⊗ π F ∗ such that ϕ = V ( v ) and (cid:107) ϕ (cid:107) = (cid:107) v (cid:107) . In particular, wehave that ( K ( E, F ) , (cid:107) · (cid:107) ) ∗ = ( E (cid:98) ⊗ π F ∗ ) / ker V . Thus, if K ( E, F ) = L ( E, F ), then D ( z )( T ) = V ( z )( T ) for every z ∈ E (cid:98) ⊗ π F ∗ and every T ∈ K ( E, F ); hence ker D = ker V and ( L ( E, F ) , (cid:107) · (cid:107) ) ∗ = ( L ( E, F ) , τ c ) ∗ . (cid:3) Let us now go towards the proof of Theorem B. We show the following result whichwill help us to prove that if (
E, F ) does not satisfy the James property, then B K ( E,F ) SOT coincides with B K ( E,F ) . Recall that the sequential closure of a set in a topological spaceis the family of all limit points of sequences on the set in consideration. Lemma 3.3.
Let E and F be Banach spaces. Suppose that there exists a norm-closedconvex set C ⊆ L ( E, F ) which is not sequentially closed in the strong operator topology.Then, ( E, F ) has the James property.Proof. Suppose that C ⊆ L ( E, F ) is norm-closed but not SOT-sequentially closed. Thisimplies that there exists a sequence of operators ( R n ) ⊆ C such that ( R n ) convergesin the SOT (and therefore in the WOT) to an operator R / ∈ C . We may (and we do)suppose that R = 0. Set K := { R n : n ∈ N } ⊂ L ( E, F ). Therefore, K is relativelyWOT-compact, 0 ∈ K W OT but 0 cannot be in co( K ) by hypothesis. Therefore, ( E, F )has the James property. (cid:3)
N THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS 7
It is not difficult to check that, for a bounded subset C of L ( E, F ), with E separable,the SOT-closure of C coincides with the SOT-sequential closure of C . Furthermore, thefollowing result shows that the unit ball of K ( E, F ) is SOT-closed if it is SOT-sequentiallyclosed under the assumption that E has the separable complementation property. Recallthat a Banach space E is said to have the separable complementation property if for everyseparable subspace Y in E , there is a separable subspace Z with Y ⊂ Z ⊂ E and Z iscomplemented in E . It is worth mentioning that D. Amir and J. Lindenstrauss provedin [2] that every weakly compactly generated Banach space (and therefore every reflexivespace) has the separable complementation property. Lemma 3.4.
Let E be a Banach space with the separable complementation property and F be an arbitrary Banach space. Then, the unit ball B K ( E,F ) is SOT-closed if and only ifit is SOT-sequentially closed.Proof. Is it enough to check that if B K ( E,F ) is not SOT-closed then it is not SOT-sequentially closed. Suppose that T is an operator which belongs to the SOT-closureof B K ( E,F ) but not to B K ( E,F ) . Notice that T is non-compact; hence there exists a separa-ble subspace E of E such that T | E is non-compact. Choose a separable subspace Z of E such that E ⊂ Z ⊂ E and Z is complemented in E . Notice that T | Z is non-compactand belongs to the SOT-closure of B K ( Z,F ) . As Z is separable, we have that T | Z is indeedin the SOT-sequential closure of B K ( Z,F ) . Let ( K n ) be a sequence in B K ( Z,F ) convergingto T | Z in the SOT. Letting P be a projection from E onto Z , it is immediate that T | Z ◦ P is non-compact and K n ◦ P is SOT-convergent to T | Z ◦ P . This proves that B K ( E,F ) is notSOT-sequentially closed. (cid:3) It is worth mentioning however that the SOT-closure and SOT-sequential closure aredifferent in general as the following remark shows.
Remark 3.5.
In general, it is not true that the SOT-sequential closure of a boundedconvex set C in L ( E, F ) coincides with the SOT-closure of C . An example is given by C := (cid:110) T ∈ B L ( (cid:96) ( ω )) : there is α < ω such that ( T ( x )) β = 0 for every β > α, x ∈ (cid:96) ( ω ) (cid:111) . It is immediate that C is SOT-sequentially closed. Nevertheless, since the canonicalprojections P α ∈ L ( (cid:96) ( ω )) with α < ω , defined by ( P α ( x )) β = x β if β (cid:54) α and 0otherwise, are in C and satisfy that { P α } α<ω SOT-converges to the identity, which is notin C , it follows that C is not SOT-closed.Notice that if E is reflexive, then it has the separable complementation property. ByLemma 3.4, B K ( E,F ) is SOT-closed if and only if it is SOT-sequentially closed. Therefore,if we assume that B K ( E,F ) SOT (cid:54) = B K ( E,F ) , then ( E, F ) has the James property by Lemma3.3. Therefore, we have the following result.
Proposition 3.6.
Let E and F be Banach spaces. If B K ( E,F ) SOT (cid:54) = B K ( E,F ) , then ( E, F ) has the James property. In order to prove Theorem B, we need the following lemma.
DANTAS, JUNG, AND MART´INEZ-CERVANTES
Lemma 3.7.
Let E be a reflexive space and F be an arbitrary Banach space. Supposethat the pair ( E, F ) has the BCAP. Then K ( E, F ) = L ( E, F ) if and only if the unit ball B K ( E,F ) is SOT-closed.Proof. First, note that as the pair (
E, F ) has the BCAP, L ( E, F ) = (cid:83) λ> λB K ( E,F ) τ c .Since B K ( E,F ) τ c ⊆ B K ( E,F ) SOT , we have that L ( E, F ) = (cid:83) λ> λB K ( E,F ) SOT . So, if weassume B K ( E,F ) to be SOT-closed, then L ( E, F ) = (cid:91) λ> λB K ( E,F ) SOT = (cid:91) λ> λB K ( E,F ) = K ( E, F ) . The other implication is immediate. (cid:3)
To prove Theorem B, we consider the following conditions and we use Theorem A,Proposition 3.1, Proposition 3.2, Proposition 3.6, and Lemma 3.7.(a) K ( E, F ) = L ( E, F ).(b) Every operator from E into F attains its norm.(c) The unit ball B K ( E,F ) is closed in the strong operator topology.(d) ( L ( E, F ) , τ c ) ∗ = ( L ( E, F ) , (cid:107)·(cid:107) ) ∗ . Proof of Theorem B.
Let E be reflexive and F be an arbitrary Banach space. It is clearthat (a) = ⇒ (b). Moreover, (b) implies that ( E, F ) does not have the James property (byapplying Theorem A), which in turn implies (c) (by applying Proposition 3.6). On theother hand, Proposition 3.2 shows (a) = ⇒ (d). By Proposition 3.1, (d) implies that ( E, F )does not have the James Property and, therefore, it implies (c) (by applying Proposition3.6). Finally, if the pair (
E, F ) has the BCAP, then the implication (c) = ⇒ (a) followsfrom Lemma 3.7. (cid:3) M.I. Ostrovskii asked in [23, §
12, pg. 65] whether there exist infinite dimensional Banachspaces on which every operator attains its norm (this question is also asked in [19, Problem8] and [14, Problem 217]). By Holub’s Theorem, if such an infinite dimensional Banachspace exists, it cannot have the AP. Theorem 3.8 below should be seen as a generalizationof this fact. Let us recall that given a (norm-closed) operator ideal A and λ (cid:62)
1, aBanach space E is said to have the λ - A -approximation property (for short, λ - A -AP) ifthe identity operator Id E belongs to { T ∈ A ( E, E ) : (cid:107) T (cid:107) (cid:54) λ } τ c . We say that E has thebounded- A -AP if it has the λ - A -AP for some λ (cid:62)
1. This general approximation propertyhas been studied, for instance, in [12, 20, 25, 28].
Theorem 3.8.
If there is an infinite dimensional Banach space E such that every operatoron L ( E ) attains its norm, then E does not have the bounded A -approximation propertyfor any ideal A not containing the identity on E .Proof. As it is highlighted in [23, §
12, pg. 66], due to a result of N.J. Kalton, if such aBanach space E exists, then it must be separable. Therefore, the SOT-closure of the set B = { T ∈ A ( E, E ) : (cid:107) T (cid:107) (cid:54) } in L ( E ) coincides with its SOT-sequential closure. Thus,if every operator on L ( E ) attains its norm, then B is SOT-closed by Lemma 3.3. Supposethat E has the bounded A -approximation property. Then, since B τ c ⊂ B SOT = B , we N THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS 9 have that B contains a multiple of the identity and therefore A contains the identity on E . (cid:3) Let us conclude the paper by showing the proof of Theorem C as a direct implicationof Theorem B and Proposition 3.9 below. Recall that a Banach space E has the Schurproperty if every weakly convergent sequence is norm convergent. It is known that aBanach space F has the Schur property if and only if every weakly compact operatorfrom E into F is compact for any Banach space E (see, for example, [27, 3.2.3, pg. 61]).Also, it is proved in [8, Theorem 1] that a Banach space F has the Schur property if andonly if the weak Grothendieck compactness principle holds in F , that is, every weaklycompact subset of F is contained in the closed convex hull of a weakly null sequence.Afterwards, W.B. Johnson et al., gave an alternative proof in [18, Theorem 1.1] for thisresult by using the Davis-Figiel-Johnsonn-Pe(cid:32)lczy´nski factorization theorem [6]. Moreover,it is observed in [18, Theorem 3.3] that a Banach space F has the Schur property if andonly if W ∞ ( E, F ) ⊂ W ( E, F ) for every Banach space E .The following result will be used as an important tool afterwards. Proposition 3.9.
Let F be a Banach space. If F fails to have the Schur property, thenthere exists a reflexive space with basis E such that K ( E, F ) (cid:54) = L ( E, F ) .Proof. Take ( x n ) ⊆ S F to be a weakly null sequence in F , which is not norm null. Since theabsolute closed convex hull of { x n : n ∈ N } is weakly compact, the operator T ∈ L ( (cid:96) , F )given by T ( e n ) := x n for each n ∈ N defines a weakly compact operator (which is notcompact). By Davis-Figiel-Johnsonn-Pe(cid:32)lczy´nski factorization theorem [6], there existsa reflexive space E such that T = S ◦ R , where R ∈ L ( (cid:96) , E ) and S ∈ L ( E , F ).In particular, notice that S cannot be a compact operator. Now, pick a weakly nullsequence ( v n ) ⊆ E so that S ( v n ) does not admit a convergent subsequence. Since ( v n ) isweakly null, consider a subsequence which is a basic sequence of E (see [1, Proposition1.5.4]) and denote it again by ( v n ). Let E := span { v n } n ∈ N . Then, E is a closed reflexivespace with basis and S ( v n ) does not admit a convergence sequence. So, we conclude that K ( E, F ) (cid:54) = L ( E, F ). (cid:3) Compared to the previously known results in [18], Theorem 3.10 below not only providesa new characterization of the Schur property in terms of norm-attaining operators, butalso shows that we can restrict the possible candidates for a domain space as in the belowitems (f)-(i) by considering only reflexive Banach spaces with basis . We refer the readerto Section 2 for the definitions of the sets V ( E, F ) and W ∞ ( E, F ). It is immediate tonotice that Theorem C follows from Theorem 3.10.
Theorem 3.10.
Let F be a Banach space. The following statements are equivalent. (a) F has the Schur property. (b) K ( E, F ) = L ( E, F ) for every reflexive space E . (c) W ∞ ( E, F ) = L ( E, F ) for every reflexive space E . (d) V ( E, F ) = L ( E, F ) for every reflexive space E . (e) NA( E, F ) = L ( E, F ) for every reflexive space E . (f) K ( G, F ) = L ( G, F ) for every reflexive space G with basis. (g) NA( G, F ) = L ( G, F ) for every reflexive space G with basis. (h) W ∞ ( G, F ) = L ( G, F ) for every reflexive space G with basis. (i) V ( G, F ) = L ( G, F ) for every reflexive space G with basis.Proof. The following diagram holds. ( c )( b ) ( d ) ( h )( a ) ( e ) ( i )( f ) ( g )Indeed, by definition we have that K ( E, F ) ⊂ W ∞ ( E, F ) ⊂ W ( E, F ) for any Banach space E and F , and it is also known that K ( E, F ) ⊂ W ∞ ( E, F ) ⊂ V ( E, F ) (see [18, Proposition3.1]). Moreover, if T is an element of V ( E, F ) with E reflexive, then T ∈ NA(
E, F ) thanksto the weak sequential compactness of B E . Thus, it is immediate that (a) = ⇒ (b) = ⇒ (c) = ⇒ (d) = ⇒ (e) = ⇒ (g) and (c) = ⇒ (h) = ⇒ (i) = ⇒ (g) hold. As a reflexive Banachspace with basis has the MAP, (f) ⇐⇒ (g) follows from Theorem B. Finally, (f) = ⇒ (a)is already obtained by Proposition 3.9. (cid:3) Acknowledgements.
We would like to thank Jos´e Rodr´ıguez for suggesting us Definition1.1 and the use of Pfitzner’s Theorem to strengthen and simplify part of the content ofthe paper. We are also grateful to Richard Aron, Gilles Godefroy, Manuel Maestre, andMiguel Mart´ın for fruitful conversations on the topic of the present paper.
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Email address : [email protected] (Jung) Department of Mathematics, POSTECH, Pohang 790-784, Republic of KoreaORCID:
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