J. Math. Anal. Appl. 413 (2014) 616–621

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Journal of Mathematical Analysis andApplications

www.elsevier.com/locate/jmaa

Operators on L∞-spaces

Kevin Beanland a,c,∗, Lon Mitchell b,c

a Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284, United Statesb Mathematical Reviews, American Mathematical Society, Ann Arbor, MI 48103, United Statesc Department of Mathematics, Washington and Lee University, Lexington, VA 24450, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 February 2013Available online 4 December 2013Submitted by Richard M. Aron

Keywords:Bounded linear operatorsL∞-space

It is shown that the space of bounded linear operators on certain L∞-spaces is non-separable. These spaces include the spaces of J. Bourgain and F. Delbaen and spacesconstructed by S.A. Argyros and R. Haydon.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

J. Bourgain and F. Delbaen developed a method (the BD-method, for short) for constructing Banach L∞-spaces [5] thathas been tremendously fruitful. They used this method to construct two classes of L∞-spaces; the first class, X, containsspaces with the Schur property, and the second, Y, contains spaces that are reflexive saturated. In 2000, R. Haydon [10]showed that the spaces in Y are each saturated by �p for some 1 < p < ∞. In 2006, S.A. Argyros and Haydon [3] usedthe BD-method in conjunction with techniques from the theory of mixed Tsirelson spaces to construct several importantBanach spaces. In particular, they constructed hereditarily indecomposable (HI) spaces whose dual is �1 and which possessthe scalar-plus-compact property. Even more recently this method has been combined with an embedding theorem ofD. Freeman, E. Odell and Th. Schlumprecht [8] to prove that every uniformly convex Banach space can be embedded into aspace with the scalar-plus-compact property [2].

In the present paper, we give a method for constructing uncountably many pairwise separated operators on certainL∞-spaces built with the BD-method. The question as to whether or not the space of operators on a given Banachspace must be non-separable has been investigated several times in the literature, e.g. by G. Androulakis et al. [1] andH.M. Wark [13]. The only known examples of spaces on which the space of operators is separable are those spaces satisfy-ing the scalar plus compact property. G. Emmanuele [7] showed that the space of operators on a space in the class X, whilenon-separable, does not contain �∞ . These seem to be the only spaces known to satisfy this property. Recall that whenevera space has an unconditional basis, the space of diagonal operators with respect to this basis is isomorphic to �∞ .

The result presented here considerably improves upon and answers a question regarding a method for constructing anon-strictly-singular operator on each of the original spaces of Bourgain and Delbaen [4]. This research is motivated by thequestion of whether or not �∞ embeds into the space of operators on certain L∞-spaces and the following theorem: If Xis an L∞-space not containing c0 and with dual isomorphic to �1, then c0 cannot be embedded into the compact operatorson X [6]. Recall that N. Kalton proved that �∞ embeds in L(X) whenever c0 does [11].

* Corresponding author at: Department of Mathematics, Washington and Lee University, Lexington, VA 24450, United States.E-mail address: beanlandk@wlu.edu (K. Beanland).

0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jmaa.2013.12.003

K. Beanland, L. Mitchell / J. Math. Anal. Appl. 413 (2014) 616–621 617

We note that M. Tarbard constructed an HI L∞-space that is similar to the Argyros–Haydon example, but which has astrictly singular non-compact operator [12]. The spaces of operators on the Tarbard and the Argyros–Haydon HI spaces areseparable.

2. Construction of some L∞-spaces

In this section we give an overview of what is now called the BD-construction method of L∞-spaces. We will follow thepresentation of Agyros and Haydon [3].

In the sequel, (�n)n∈N denotes a sequence of finite sets. Let Γn = ⋃k�n �k and Γ = ⋃

n∈N �n . Let (eγ )γ ∈Γ denotethe unit coordinate vectors in �∞(Γ ), i.e. eγ (δ) = 1 if δ = γ and 0 otherwise, and let (e∗

γ )γ ∈Γ denote the associatedbiorthogonal functionals.

The basis of the BD-space will be a sequence of vectors in �∞(Γ ) denoted by (dγ )γ ∈Γ ; let (d∗γ )γ ∈Γ denote the biorthog-

onal functionals. Exact conditions for when (d∗γ )γ ∈Γ is a basis for �1(Γ ) are given by Argyros and Haydon [3, Theorem 3.5].

Let D∗(0,k] be the associated basis projection:

D∗(0,k]d

∗γ =

{d∗γ γ ∈ Γk,

0 γ ∈ Γ \ Γk.

For an interval of integers I let D∗I denote the natural projection onto the coordinates in

⋃i∈I �i = �I .

The vectors in Γ will vary among the different L∞-spaces we wish to investigate, but in each case �1 is a single-elementset and the finite sets �n are defined recursively. To obtain a unifying view of the different spaces, since Γ is merely anindex set, we will choose to equate each γ in a particular Γ with a six-tuple, and the data encoded in each six-tuple willallow us to capture details of the corresponding recursive construction. The generic form will be

γ = (rank(γ ),bγ ,base(γ ),age(γ ), w(γ ), top(γ )

),

with restrictions on the entries as follows:

• Rank: rank(γ ) = n if and only if γ ∈ �n .• Base coefficient: bγ ∈ (0,1].• Base: Either base(γ ) ∈ Γk for some k < rank(γ ) − 1 or base(γ ) = ∅. In particular, if rank(γ ) � 2, then base(γ ) = ∅, but

the reverse implication need not hold.• Age: age(γ ) ∈N.• Weight: w(γ ) ∈ [0,1].• Top vector: Let Iγ = (rank(base(γ )), rank(γ )) ∩N, using rank(∅) = 0. If Iγ = ∅, then top(γ ) = 0. Otherwise,

top(γ ) = D∗Iγ

∑η∈�Iγ

aγη e∗

η

for some scalar sequence (aγη )η∈�Iγ

satisfying∑

η∈�Iγ|aγ

η | � 1.

Any Γ that can be described in this way we will call a BD-index set, and our main results pertain to such Γ .We now outline some basic properties of the vectors d∗

γ : For n ∈ N and γ ∈ �n ,

d∗γ = e∗

γ − c∗γ

where c∗γ is a vector supported in �[1,n) . In particular, the vector dγ is supported on Γ \ Γrank(γ )−1 and its support in

Γrank(γ ) is exactly eγ [3], so that if rank(γ ) < rank(η), then e∗γ (dη) = 0, and if rank(γ ) = rank(η) but γ �= η, then e∗

γ (dη) = 0.For γ ∈ �n , the following decomposition holds:

e∗γ = d∗

γ + w(γ )D∗Iγ

∑η∈�Iγ

aγη e∗

η + bγ e∗base(γ ). (1)

By recursively applying the above formula one can arrive at a formula for e∗γ in terms of (d∗

γ )γ ∈Γ .As observed in [3], the BD-index set admits a natural tree structure that yields a decomposition of the norming function-

als (e∗γ )γ ∈Γ in terms of the functionals (d∗

γ )γ ∈Γ . Indeed, this tree structure follows from the fact that the ‘base’ and ‘top’of each tuple have strictly smaller ranks than the tuple itself. For the purposes of the present paper, since we will not needthe tree decompositions of the functionals, we omit this description and refer the interested reader to the papers [3,9].

Let X(Γ ) denote the subspace of �∞(Γ ) that is the closed linear span of (dγ )γ ∈Γ . We are interested in the case when(dγ )γ ∈Γ is a Schauder basis for X(Γ ). For x ∈ X(Γ ),

‖x‖X(Γ ) = supγ ∈Γ

∣∣e∗γ (x)

∣∣.

618 K. Beanland, L. Mitchell / J. Math. Anal. Appl. 413 (2014) 616–621

3. Operators on BD-constructions

We now state a sufficient condition for a permutation of a BD-index set Γ to give rise to a bounded operator on X(Γ ).

Proposition 1. Let π be a permutation of Γ such that π(�n) = �n for all n ∈ N (we say that the permutation is rank preserving) andsuppose (dγ )γ ∈Γ is a basis for X(Γ ). If there is a k ∈ N such that π |Γk is the identity and e∗

π(γ )(dπ(ξ)) = e∗γ (dξ ) for ξ ∈ Γ \ Γk and

γ ∈ Γ , then the map Tπ defined by setting Tπ (dγ ) = dπ(γ ) and extending linearly is bounded.

Proof. Since Tπ is linear, it is bounded when restricted to any finite dimensional space. Thus it suffices to show that Tπ

is an isometry when restricted to the finite co-dimensional space with basis (dγ )γ ∈Γ \Γk . Let (cγ )γ ∈Γ be an arbitrary scalarsequence. Then

∥∥∥∥∥∞∑

n=k+1

∑γ ∈�n

cγ dπ(γ )

∥∥∥∥∥ = supη∈Γ

∣∣∣∣∣∞∑

n=k+1

∑γ ∈�n

cγ e∗π(η)(dπ(γ ))

∣∣∣∣∣

= supη∈Γ

∣∣∣∣∣∞∑

n=k+1

∑γ ∈�n

cγ e∗η(dγ )

∣∣∣∣∣

=∥∥∥∥∥

∞∑n=k+1

∑γ ∈�n

cγ dγ

∥∥∥∥∥.

This proves the claim. �Definition 2. For γ , ξ ∈ Γ we write γ ≈ ξ if rank(γ ) = rank(ξ), bγ = bξ , w(γ ) = w(ξ), age(γ ) = age(ξ) and rank(base(γ )) =rank(base(ξ)).

Now we give a sufficient condition for a permutation to satisfy the assumptions of Proposition 1.

Proposition 3. Let n ∈ N and suppose Γ is a BD-index set such that there exists a permutation π : Γ → Γ satisfying

(i) π(γ ) ≈ γ and aπ(γ )η = aγ

π−1(η)for γ ∈ Γ and η ∈ Iγ ,

(ii) π(base(γ )) = base(π(γ )) for all γ ∈ Γ with rank(base(γ )) > n, and(iii) π |Γn is the identity.

Then π satisfies the assumptions of Proposition 1 and the operator Tπ is bounded on X(Γ ).

Note that since π(γ ) ≈ γ , we have Iγ = Iπ(γ ) . Also note that aπ(γ )η = aγ

π−1(η)implies

∑η∈�Iπ(γ )

aπ(γ )η e∗

η =∑

η∈�Iγ

aγη e∗

π(η). (2)

Using Proposition 1 it suffices to show that for ξ ∈ Γ \ Γn and γ ∈ Γ we have

e∗π(γ )(dπ(ξ)) = e∗

γ (dξ ). (3)

For rank(γ ) < rank(ξ), we have e∗π(γ )(dπ(ξ)) = e∗

γ (dξ ) = 0. Let q = rank(ξ). We will show that for all i � 0 and γ ∈ �q+i ,Eq. (3) holds. We proceed by induction on i � 0. For the base case we have

e∗π(γ )(dπ(ξ)) = d∗

π(γ )(dπ(ξ)) = d∗γ (dξ ) = e∗

γ (dξ ).

For the inductive step, let i ∈N and assume that for 0 � k < i and γ ∈ �q+k , Eq. (3) holds. Let γ ∈ �q+i . Using our inductionhypothesis, Eq. (2), and the fact that π(γ ) ≈ γ , we have

e∗π(γ )(dπ(ξ)) = d∗

π(γ )(dπ(ξ)) + w(π(γ )

)(D∗

Iπ(γ )

∑η∈�Iπ(γ )

aπ(γ )η e∗

η

)(dπ(ξ)) + bπ(γ )e∗

base(π(γ ))(dπ(ξ))

= d∗γ (dξ ) + w(γ )

∑η∈�I

aγη e∗

π(η)(D Iγ dπ(ξ)) + bγ e∗base(π(γ ))(dπ(ξ)).

γ

K. Beanland, L. Mitchell / J. Math. Anal. Appl. 413 (2014) 616–621 619

By our induction hypothesis and the definition of D Iγ we have that for η ∈ �Iγ , e∗π(η)(D Iπ(γ )

dπ(ξ)) = e∗η(D Iγ dξ ). Also note

that

e∗base(π(γ ))(dπ(ξ)) = e∗

π(base(γ ))(dπ(ξ)) = e∗base(γ )(dξ ).

To see the first equality: when rank(base(γ )) > n apply assumption (ii) and when rank(base(γ )) � n observe that both sidesare 0. The second equality follows from the induction hypothesis. Using these observations and the induction hypothesis wecontinue as follows

e∗π(γ )(dπ(ξ)) = d∗

γ (dξ ) + w(γ )∑

η∈�Iγ

aγη e∗

η(D Iγ dξ ) + bγ e∗base(γ )(dξ )

= d∗γ (dξ ) + w(γ )

(D∗

Iγ

∑η∈�Iγ

aγη e∗

η

)(dξ ) + bγ e∗

base(γ )(dξ )

= e∗γ (dξ ).

This concludes the proof. �Definition 4. A BD-index set Γ is called symmetric if:

• For every rank preserving permutation p of Γ , γ ∈ Γ and ξ ≈ base(γ )(rank(γ ),bγ , ξ,age(γ ), w(γ ), D∗

Iγ

∑η∈�Iγ

aγη e∗

p(η)

)∈ Γ.

• There exist positive integers k and n, with 1 < k < n − 1, and elements γ �= γ̃ in �k , such that, for each i > n, there

exist γi �= γ̃i in �i satisfying γi ≈ γ̃i , aγiη = aγ̃i

η for η ∈ �Iγi= �Iγ̃i

, base(γi) = γ , and base(γ̃i) = γ̃ .

Consider the following uncountable index set:

D = {e = (ε j) j∈N: ε j = 0 for all j � n and ε j ∈ {0,1} for all j > n

}.

Note that the final assumption is stronger than assuming that there exist nodes of arbitrarily large rank that only disagreeon their bases.

Proposition 5. If Γ is symmetric, then there are uncountably many permutations (πe)e∈D such that (Tπe )e∈D ⊂ L(X(Γ )) and fordistinct e,h ∈ D, ‖Tπe − Tπh‖� 1/C where C is the basis constant of (dγ )γ ∈Γ .

Proof. Since Γ is symmetric, there exist k,n ∈N with 1 < k < n − 1 and γ0 �= γ̃0 in �k such that for each i > n we can find

γi �= γ̃i in �i with γi ≈ γ̃i , base(γi) = γ0, base(γ̃i) = γ̃0 and aγiη = aγ̃i

η for η ∈ �Iγi= �Iγ̃i

.We generate the uncountable number of permutations by either switching or not switching the elements γi and γ̃i in

each �i with i > n, and thus condition (iii) of Proposition 3 will automatically be satisfied.For each e ∈ D define πe inductively as follows. Let i0 = min{i: εi = 1}.

• Let πe|Γi0 \{γi0 ,γ̃i0 } be the identity.• Let πe(γi0 ) = γ̃i0 and πe(γ̃i0 ) = γi0 .• To define πe(γ ) for γ ∈ Γ \ Γi0 , suppose that πe|Γi has been defined for i � i0.

(a) If εi+1 = 0 then for γ ∈ �i+1 let

πe(γ ) =(

i + 1,bγ ,πe(base(γ )

), w(γ ),age(γ ), D∗

Iγ

∑η∈�Iγ

aγη e∗

πe(η)

).

(b) If εi+1 = 1 then let πe(γi+1) = γ̃i+1 and πe(γ̃i+1) = γi+1, and for γ ∈ �i+1 \ {γi, γ̃i} let

πe(γ ) =(

i + 1,bγ ,πe(base(γ )

), w(γ ),age(γ ), D∗

Iγ

∑η∈�Iγ

aγη e∗

πe(η)

).

We must show that πe is well-defined. In fact, πe is a permutation of Γ of order 2 such that πe(�i) ⊆ �i for each i. Forthis we use that Γ is symmetric. It is clear from the definition that πe(base(γ )) ≈ base(γ ) for all γ ∈ Γi0 (since πe is theidentity and γi0 ≈ γ̃i0 ), which implies that πe is well-defined on Γi0 and that π(�i) ⊆ �i for all i � i0, and it is further clear

620 K. Beanland, L. Mitchell / J. Math. Anal. Appl. 413 (2014) 616–621

that πe is a permutation of order 2 on Γi0 . By construction, if πe satisfies these properties on some Γi , then πe will havethe same properties on Γi+1. Thus, by induction, πe(base(γ )) ≈ base(γ ) for all γ ∈ Γ and πe is a well-defined permutationof order 2 with πe(�i) ⊆ �i for each i.

It remains to prove that conditions (i) and (ii) of Proposition 3 hold. Using the definition, γ ≈ πe(γ ) and aπ(γ )η = aγ

π−1(η)

for γ ∈ Γ and η ∈ Iγ . So, (i) holds. Item (ii) follows directly from the definition of πe . Note that base(γi),base(γ̃i) ∈ �k andk < n.

The fact that (Tπe )e∈D ⊂ L(X(Γ )) follows from Proposition 3. Let h �= e in D and find j ∈ {i: εi �= hi}. Assume withoutloss of generality that ε j = 0 and h j = 1. Therefore πh(γ j) = γ̃ j . Since base(γ̃ j) = γ̃0 and base(πe(γ j)) = πe(base(γ j)) =πe(γ0) = γ0, it follows that πe(γ j) �= πh(γ j). Therefore Tπe dγ j = dπe(γ j) �= dγ̃ j

= Tπh dγ j .Thus

‖Tπe − Tπh‖� ‖Tπedγ j − Tπhdγ j ‖ = ‖dπe(γ j) − dγ̃ j‖� 1

C

where C is the basic constant of (dγ )γ ∈Γ . �Theorem 6. If Γ is symmetric then L(X(Γ )) is non-separable.

Remark 1. Since Tπe is the identity on many elements it is easy to see that no sequence taken from (Tπe )e∈D is equivalentto c0.

4. Examples

In this section we prove that several BD-index sets from the literature are symmetric. To recall the constructions, wesimply specify the allowable tuples

γ =(

n,bγ ,base(γ ),age(γ ), w(γ ), D∗Iγ

∑η∈�Iγ

aγη e∗

η

).

Recall that the allowable base elements base(γ ) are always chosen with rank strictly less than one minus the rank of γ(and can be the emptyset) and

Iγ = (rank

(base(γ )

), rank(γ )

) ∩N.

Example 1 (The original BD-spaces). For spaces in class Y, the index set Γ Y is defined as follows: Fix α and β such that0 < β < α < 1 with α + β > 1 and α + 2βλ � λ for some λ > 1. For n ∈ N, γ ∈ �n is of the form

γ = (n,α,base(γ ),1, β,±D∗

Iγ e∗η

)for η ∈ �Iγ . It is easy to see that Γ Y is symmetric.

Example 2 (The spaces of Gasparis, Papadiamantis and Zisimopoulou). I. Gasparis, M.K. Papadiamantis and D.Z. Zisimopoulou [9]defined a class of L∞ spaces that are �p saturated (for some p ∈ (1,∞)).

Let n ∈ N and (βk)nk=1 be such that β1 < 1, βk < 1/2 for all k ∈ {2, . . . ,n} and

∑nk=1 βk > 1. For n ∈ N, γ ∈ �n is of the

form

γ = (n, β1,base(γ ),2, β2,±D∗

Iγ e∗η

)for η ∈ �Iγ or

γ = (n,1,base(γ ),k, βk,±D∗

Iγ e∗η

)for η ∈ �Iγ and any k > 2. In the above we have the restriction that base(γ ) is chosen such that w(base(γ )) = βk−1.

We will show that Γ is symmetric. The first requirement follows from the definition. For the second, let k,n ∈ N with0 < k < n, γ �= γ̃ in �k and η ∈ �n . The nodes γi and γ̃i , i > n, can be defined as

γi = (i, β1, γ ,2, β2, D∗

(k,n+1)e∗η

)and

γ̃i = (i, β1, γ̃ ,2, β2, D∗

(k,n+1)e∗η

).

Note that D∗(k,n+1)

e∗η = D∗

(k,i)e∗η for i > n.

K. Beanland, L. Mitchell / J. Math. Anal. Appl. 413 (2014) 616–621 621

Example 3 (The BD-spaces of Argyros and Haydon). Let (m j) j∈N and (n j) j∈N be increasing sequences of natural numbers.Argyros and Haydon defined a BD-space BmT [(m j,n j)]. The nodes γ ∈ �n have the following form:

γ =(

n,1,base(γ ),age(γ ),m−1j , D∗

Iγ

∑η∈�Iγ

aγη e∗

η

)

where j � n, if base(γ ) �= ∅ then w(base(γ )) = m−1j , age(base(γ )) = age(γ ) − 1 and

∑η∈�Iγ

|aγη | � 1 where each aη is

rational with denominator dividing Nn! for some fixed increasing sequence (Nn)n∈N . Also, age(γ ) = 0 when base(γ ) = ∅.To see that the index set is symmetric, choose k and n with 0 < k < n, choose γ , γ̃ ∈ �k with w(γ ) = w(γ̃ ) = m−1

1 andage(γ ) = age(γ̃ ) = 0, and let η ∈ �n . The nodes γi and γ̃i , i > n, can be defined as

γi = (i,1, γ ,1,m−1

1 , D∗(k,n+1)e

∗η

)and

γ̃i = (i,1, γ̃ ,1,m−1

1 , D∗(k,n+1)e

∗η

).

Note that D∗(k,n+1)

e∗η = D∗

(k,i)e∗η for i > n.

Example 4 (The scalar-plus-compact space of Argyros and Haydon). Let Γ K denote the index set of the scalar-plus-compactspace XK of Argyros and Haydon. As a final example, we show that for e ∈ D (not all zeros) the permutation πe is not welldefined on Γ K ; thus the corresponding operator is not well defined. This must be the case since L(XK ) is separable.

As in the previous example, there are fixed increasing sequences of natural numbers (n j) j∈N and (m j) j∈N satisfyingcertain growth conditions. We will not give the exact specifications for the construction of Γ K . The definition of Γ K usesan injective function σ : Γ K → N satisfying certain growth conditions. Note that by definition w(γ ) = w(πe(γ )). Let δ1 �= δ2in Γ K with πe(δ1) = δ2 and πe(δ2) = δ1. Let n ∈N be large enough so that the tuples

ξ1 = (n,1,∅,1,m−1

2 j−1, e∗δ1

)and ξ2 = (

n,1,∅,1,m−12 j−1, e∗

δ2

)are in Γ K . By definition πe(ξ1) = ξ2 and πe(ξ2) = ξ1. Let m ∈N and η ∈ �(n,m) such that

α = (m,1, ξ1,1,m−1

2 j−1, D∗(n,m)e

∗η

)is in Γ K . By the definition of the injective function σ (see [3, page 15]), w(η) = m−1

4(σ (ξ1))�= m−1

4(σ (πe(ξ1)))= m−1

4(σ (ξ2)). We

claim that πe(α) /∈ Γ K . By definition

πe(α) = (m,1, ξ2,1,m−1

2 j−1, D∗(n,m)e

∗πe(η)

).

As a consequence of the definition of Γ K , in order to have πe(α) ∈ Γ K , we must have w(πe(η)) = m−14(σ (ξ2)) . However

w(πe(η)) = w(η) �= m−14(σ (ξ2)) , as seen above.

Acknowledgment

The authors wish to thank the anonymous referee for a superb job resulting in many improvements to this paper.

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