Subspaces that can and cannot be the kernel of a bounded operator on a Banach space
aa r X i v : . [ m a t h . F A ] N ov SUBSPACES THAT CAN AND CANNOT BE THE KERNEL OF ABOUNDED OPERATOR ON A BANACH SPACE
NIELS JAKOB LAUSTSEN AND JARED T. WHITE
Abstract.
Given a Banach space E , we ask which closed subspaces may be realised asthe kernel of a bounded operator E Ñ E . We prove some positive results which imply inparticular that when E is separable every closed subspace is a kernel. Moreover, we showthat there exists a Banach space E which contains a closed subspace that cannot be realisedas the kernel of any bounded operator on E . This implies that the Banach algebra B p E q ofbounded operators on E fails to be weak*-topologically left Noetherian in the sense of [7].The Banach space E that we use is the dual of one of Wark’s non-separable, reflexive Banachspaces with few operators. Introduction
In this note we address the following natural question: given a Banach space E , which ofits closed linear subspaces F are the kernel of some bounded linear operator E Ñ E ? Weshall begin by showing that if either E { F is separable, or F is separable and E has theseparable complementation property, then F is indeed the kernel of some bounded operatoron E (Propositions 2.1 and 2.2). Our main result is that there exists a reflexive, non-separableBanach space E for which these are the only closed linear subspaces that may be realisedas kernels (Theorem 2.6), and in particular E has a closed linear subspace that cannot berealised as the kernel of a bounded linear operator on E (Corollary 2.7). The Banach space inquestion may be taken to be the dual of any reflexive, non-separable Banach space that hasfew operators, in the sense that every bounded operator on E is the sum of a scalar multiple ofthe identity and an operator with separable range. Wark has shown that such Banach spacesexist [5, 6].We now describe how we came to consider this question. Given a Banach space E we write E for its dual space, and B p E q for the algebra of bounded linear operators E Ñ E . We recallthat a dual Banach algebra is a Banach algebra A which is isomorphically a dual Banach spacein such a way that the multiplication on A is separately weak*-continuous; equivalently A hasa predual which may be identified with a closed A -submodule of A . When E is a reflexiveBanach space, B p E q is a dual Banach algebra with predual given by E p b E , where p b denotesthe projective tensor product of Banach spaces. We recall the following definition from [7]: Definition 1.1.
Let A be a dual Banach algebra. We say that A is weak*-topologically leftNoetherian if every weak*-closed left ideal I of A is weak*-topologically finitely-generated, i.e.there exists n P N , and there exist x , . . . , x n P I such that I “ Ax ` C x ` ¨ ¨ ¨ ` Ax n ` C x nw ˚ . Date : 2018.2010
Mathematics Subject Classification.
Primary: 46H10, 47L10; secondary: 16P40, 46B26, 47A05, 47L45.
Key words and phrases.
Banach space, bounded operator, kernel, dual Banach algebra, weak*-closed ideal,Noetherian.
In [7] various examples were given of dual Banach algebras which satisfy this condition,but none were found that fail it. Using our main result, we are able to prove in Theorem 2.8of this note that, for any non-separable, reflexive Banach space E with few operators in theabove sense, B p E q is a dual Banach algebra which is not weak*-topologically left Noetherian.2. Results
We first show that in many cases closed linear subspaces can be realised as kernels. In par-ticular, for a separable Banach space every closed linear subspace is the kernel of a boundedlinear operator. Given a Banach space E , and elements x P E, λ P E , we use the bra-ketnotation | x yx λ | to denote the rank-one operator y ÞÑ x y, λ y x . Proposition 2.1.
Let E be a Banach space, and let F be a closed linear subspace of E suchthat E { F is separable. Then there exists T P B p E q such that ker T “ F .Proof. Since E { F is separable, the unit ball of F K – p E { F q is weak*-metrisable, and hence,since it is also compact, it is separable. Therefore we may choose a sequence of functionals p λ n q which is weak*-dense in the unit ball of F K . We may assume that E is infinite dimensional,since otherwise the result follows from elementary linear algebra. We may therefore pick anormalised basic sequence p b n q in E . Define T P B p E q by T “ ÿ n “ ´ n | b n yx λ n | . Since each λ n belongs to F K , clearly F Ă ker T . Conversely, if x P ker T then, since p b n q is abasic sequence, we must have x x, λ n y “ for all n P N . Hence x P t λ n : n P N u K “ ´ span w ˚ t λ n : n P N u ¯ K “ p F K q K “ F, as required. (cid:3) A Banach space E is said to have the separable complementation property if, for eachseparable linear subspace F of E , there is a separable, complemented linear subspace D of E such that F Ă D . For such Banach spaces we can show that every separable closed linearsubspace is a kernel. By [2] every reflexive Banach space has the separable complementationproperty, so that the next proposition applies in particular to the duals of Wark’s Banachspaces, which we shall use in our main theorems. We refer to [1] for a survey of more generalclasses of Banach spaces that enjoy the separable complementation property. Proposition 2.2.
Let E be a Banach space with the separable complementation property.Then, for every closed, separable linear subspace F of E , there exists T P B p E q such that ker T “ F .Proof. Choose a separable, complemented linear subspace D of E such that F Ă D , and let P P B p E q be a projection with range D . By Proposition 2.1, we can find S P B p D q such that ker S “ F . Then T : x ÞÑ SP x ` x ´ P x, E Ñ E, defines a bounded linear operator on E . We shall now complete the proof by showing that ker T “ F .Indeed, for each x P ker T we have “ p id E ´ P q T x “ x ´ P x , so that
P x “ x . This impliesthat “ T x “ Sx , and therefore x P F . Conversely, each x P F satisfies Sx “ and P x “ x ,from which it follows that T x “ . (cid:3) UBSPACES AS KERNELS OF BOUNDED OPERATORS 3
We recall some notions from Banach space theory that we shall require. Let E be a Banachspace. A biorthogonal system in E is a set tp x γ , λ γ q : γ P Γ u Ă E ˆ E , for some indexing set Γ , with the property that x x α , λ β y “ if α “ β otherwise p α, β P Γ q . A Markushevich basis for a Banach space E is a biorthogonal system tp x γ , λ γ q : γ P Γ u in E such that t λ γ : γ P Γ u separates the points of E and such that span t x γ : γ P Γ u “ E . For anin-depth discussion of Markushevich bases see [1], in which a Markushevich basis is referredto as an “M-basis”.We now prove a lemma and its corollary which we shall use to prove Corollary 2.7 below. Lemma 2.3.
Let E be a Banach space containing an uncountable biorthogonal system. Then E contains a closed linear subspace F such that both F and E { F are non-separable.Proof. Let tp x γ , λ γ q : γ P Γ u be an uncountable biorthogonal system in E . We can write Γ “ Ť n “ Γ n , where Γ n “ t γ P Γ : } x γ } , } λ γ } ď n u p n P N q . Since Γ is uncountable, there must exist an n P N such that Γ n is uncountable. Let ∆ be anuncountable subset of Γ n such that Γ n z ∆ is also uncountable, and set F “ span t x γ : γ P ∆ u .The subspace F is non-separable since t x γ : γ P ∆ u is an uncountable set satisfying } x α ´ x β } ě n |x x α ´ x β , λ α y| “ n p α, β P ∆ , α ‰ β q . Let q : E Ñ E { F denote the quotient map. It is well known that the dual map q : p E { F q Ñ E is an isometry with image equal to F K . For each γ P Γ n z ∆ the functional λ γ clearly belongsto F K , so that there exists g γ P p E { F q such that q p g γ q “ λ γ , and such that } g γ } “ } λ γ } . Wenow see that t q p x γ q : γ P Γ n z ∆ u is an uncountable { n -separated subset of E { F because } q p x α q ´ q p x β q} ě n |x q p x α q ´ q p x β q , g α y| “ n |x x α ´ x β , q p g α qy|“ n |x x α ´ x β , λ α y| “ n p α, β P Γ n z ∆ , α ‰ β q . It follows that E { F is non-separable. (cid:3) Corollary 2.4.
Let E be a non-separable, reflexive Banach space. Then E contains a closedlinear subspace F such that both F and E { F are non-separable.Proof. By [1, Theorem 5.1] E has a Markushevich basis tp x γ , λ γ q : γ P Γ u . The set Γ mustbe uncountable since E is non-separable and, by the definition of a Markushevich basis, span t x γ : γ P Γ u “ E . Hence the result follows from Lemma 2.3. (cid:3) We now move on to discuss our main example. Building on the work of Shelah andStepr¯ans [4], Wark constructed in [5] a reflexive Banach space E W with the property thatit is non-separable but has few operators in the sense that(2.1) B p E W q “ C id E W ` X p E W q , where X p E W q denotes the ideal of operators on E W with separable range. Recently Warkgave a second example of such a space [6] with the additional property that the space is N. J. LAUSTSEN AND J. T. WHITE uniformly convex. For the rest of our paper E W can be taken to be either of these spaces.In particular, the only properties of E W that we shall make use of are that it is reflexive,non-separable, and satisfies Equation (2.1). Remark.
We briefly outline why the dual Banach algebra B p E W q fits into the frameworkof [7]. A transfinite basis for a Banach space X is a linearly independent family t x α : α ă γ u of vectors in X , where γ is some infinite ordinal, such that X “ span t x α : α ă γ u is densein X , and with the property that there is a constant C ě such that, for each ordinal β ă γ ,the linear map P β : X Ñ X defined by P β ´ ÿ α ă γ s α x α ¯ “ ÿ α ă β s α x α has norm at most C .In the notation of [5] and [6] the family t e p α q : α ă ω u is a transfinite basis of E W . Seethe proofs of Theorem 2 in [5] or Proposition 8 in [6]. It is shown in [3] that Banach spaceswith transfinite bases have the approximation property. Since the duals of reflexive Banachspaces with the approximation property also have this property, E W has the approximationproperty. It follows from [7, Corollary 5.6] that the algebra of compact operators K p E W q is acompliant Banach algebra in the sense of [7, Definition 5.4]. Hence K p E W q , and its multiplieralgebra B p E W q , fit into the framework of that paper. In particular [7, Theorem 6.3] givesa complete description of the weak*-closed left ideals of B p E W q , although we shall not needthis in the sequel.In what follows we denote the image of a bounded linear operator T by im T . Proposition 2.5.
Let F be a closed linear subspace of the Banach space E W with the propertythat F “ im T ` ¨ ¨ ¨ ` im T n , for some n P N , and T , . . . , T n P B p E W q . Then either F or E W { F is separable.Proof. Suppose that F “ im T ` ¨ ¨ ¨ ` im T n , for some n P N , and some T , . . . , T n P B p E W q .By (2.1) there exist α , . . . , α n P C and S , . . . , S n P X p E W q such that T i “ α i id E W ` S i p i “ , . . . , n q . If every α i equals zero, then F “ im S ` ¨ ¨ ¨ ` im S n , which is separable. Otherwise, withoutloss of generality, we may assume that α ‰ . Let x P E W . Then T x “ α x ` S x , implyingthat x “ α p T x ´ S x q P F ` im S . As x was arbitrary, it follows that E W “ F ` im S , so that E W { F “ ` F ` im S ˘ F – im S ` im S X F ˘ . Hence, it follows that E W { F is separable. (cid:3) We can now prove our two theorems.
Theorem 2.6.
Let D be a closed linear subspace of E W . Then the following conditions areequivalent: (a) either D or E W { D is separable; (b) D “ ker T , for some T P B p E W q ; UBSPACES AS KERNELS OF BOUNDED OPERATORS 5 (c) there exist n P N , and T , . . . , T n P B p E W q such that D “ Ş ni “ ker T i .Proof. We first prove that (a) implies (b). Indeed, let D be a closed linear subspace of E W .By [2], we may apply Proposition 2.2 to E W to see that if D is separable then it may berealised as the kernel of some T P B p E W q . If instead E W { D is separable, then we may applyProposition 2.1.It is trivial that (b) implies (c), so it remains to prove that (c) implies (a). Let D be aclosed linear subspace of E W that can be written in the given form for some n P N and some T , . . . , T n P B p E W q . Set F “ D K . Since E W is reflexive, there exist S , . . . , S n P B p E W q such that, for each i “ , . . . , n , T i “ S i , the dual operator of S i . It follows that F “ D K “ ˜ n č i “ ker S i ¸ K “ im S ` ¨ ¨ ¨ ` im S n . Hence, by Proposition 2.5, either F or E W { F is separable. Since F is also reflexive, it nowfollows from the formulae p E W { F q – D and F – E W { D that either D or E W { D is separable. (cid:3) Corollary 2.7.
The Banach space E W contains a closed linear subspace D which is not ofthe form Ş ni “ ker T i , for any choice of n P N , and operators T , . . . , T n P B p E W q .Proof. By Corollary 2.4 E W contains a closed linear subspace D such that neither D nor E W { D is separable. The result now follows from Theorem 2.6. (cid:3) Theorem 2.8.
The dual Banach algebra B p E W q is not weak*-topologically left Noetherian.Proof. To simplify notation set E “ E W . Let D be a closed linear subspace of E as inCorollary 2.7 and set I : “ t T P B p E q : ker T Ą D u . It is clear that I is a left ideal of B p E q , and it is weak*-closed since I “ x b λ : x P D, λ P E ( K . We shall show that this ideal fails to be weak*-topologically finitely-generated. Assume to-wards a contradiction that there exist n P N and T , . . . , T n P B p E q such that I “ B p E q T ` ¨ ¨ ¨ ` B p E q T nw ˚ . We show that č T P I ker T “ n č i “ ker T i . Indeed, let x P Ş ni “ ker T i , and S P I . Take a net p S α q in B p E q T ` ¨ ¨ ¨ ` B p E q T n convergingto S in the weak*-topology. Then for any λ P E we have “ lim α x S α p x q , λ y “ lim α x x b λ, S α y “ x x b λ, S y “ x S p x q , λ y , and as λ was arbitrary it follows that S p x q “ . As x was arbitrary Ş ni “ ker T i Ă Ş T P I ker T ,and the reverse inclusion is trivial.Observe that D Ă Ş T P I ker T . Conversely, given x P E z D , we may pick λ P E such that x x, λ y “ , and ker λ Ą D . Then the operator | x yx λ | belongs to I , but | x yx λ |p x q “ x ‰ , sothat in fact D “ Ş T P I ker T “ Ş ni “ ker T i . However this contradicts the choice of D . (cid:3) N. J. LAUSTSEN AND J. T. WHITE
This is the only example that we know of a dual Banach algebra which is not weak*-topologically left Noetherian. It would be interesting to know if there are examples of the form M p G q , the measure algebra of a locally compact group G . In the light of [7, Corollary 1.6(i)]this would be particularly interesting for a compact group G . It would also be interestingto know whether the Fourier–Stieltjes algebra of a locally compact group ever fails to beweak*-topologically left Noetherian.Another interesting problem would be to characterise those closed linear subspaces F of anon-separable Banach space E such that F is the kernel of some bounded linear operator on E . Acknowledgements.
The second author is supported by the French “Investissements d’Av-enir” program, project ISITE-BFC (contract ANR-15-IDEX-03). The article is based on partof the second author’s PhD thesis, and as such we would like to thank Garth Dales, as wellas the thesis examiners Gordon Blower and Tom Körner, for their careful reading of earlierversions of this material and their helpful comments. We are grateful to Hugh Wark andTomasz Kania for some useful email exchanges.
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E-mail address : [email protected] Jared T. White, Laboratoire de Mathématiques de Besançon, Université de Franche-Comté,16 Route de Gray, 25030 Besançon, France.
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