On the permutative equivalence of squares of unconditional bases
aa r X i v : . [ m a t h . F A ] F e b ON THE PERMUTATIVE EQUIVALENCE OFSQUARES OF UNCONDITIONAL BASES
F. ALBIAC AND J. L. ANSORENA
Abstract.
We prove that if the squares of two unconditionalbases are equivalent up to a permutation, then the bases themselvesare permutatively equivalent. This settles a twenty year-old ques-tion raised by Casazza and Kalton in [13]. Solving this problemprovides a new paradigm to study the uniqueness of unconditionalbasis in the general framework of quasi-Banach spaces. Multipleexamples are given to illustrate how to put in practice this theo-retical scheme. Among the main applications of this principle weobtain the uniqueness of unconditional basis up to permutation offinite sums of quasi-Banach spaces with this property. Introduction and background
An important long-standing problem in Banach space theory, eventu-ally solved in the negative by Gowers and Maurey in 1997 [18], askedwhether any two Banach spaces X and Y such that X is isomorphicto a complemented subspace of Y and such that Y is isomorphic to acomplemented subspace of X are isomorphic. This is known, by anal-ogy with a similar result for cardinals in the category of sets, as the Schr¨oder-Bernstein problem for Banach spaces.Pe lczy´nski had noticed much earlier, back in 1969, that a little ex-tra information about each space, namely being isomorphic to theirsquares, is all that is needed for the Schr¨oder-Bernstein problem forBanach spaces to have a positive outcome [31]. This observation, nowa-days known as
Pe lczy´nski’s decomposition method , highlighted the roleplayed by the squares of the spaces, and the question arose whether
Mathematics Subject Classification.
Key words and phrases. uniqueness, unconditional basis, equivalence of bases,quasi-Banach space, Banach lattice.Both authors supported by the Spanish Ministry for Science, Innovation, andUniversities, Grant PGC2018-095366-B-I00 for
An´alisis Vectorial, Multilineal y Ap-proximaci´on . The first-named author also acknowledges the support from SpanishMinistry for Economy and Competitivity, Grant MTM2016-76808-P for
Operators,lattices, and structure of Banach spaces . any two Banach spaces X and Y such that X ≈ Y are isomorphic.This problem was also settled in the aforementioned article by Gowersand Maurey. Indeed, the authors constructed in [18] a Banach space X with X ≈ X but X X . Then, if we put Y = X , we have that X is isomorphic to a complemented subspace of Y , that Y is isomorphicto a subspace of X , that X ≈ Y , and that X Y . So, the pair ofspaces X and Y serves as a counterexample for both questions.The Schr¨oder-Bernstein problem for Banach spaces is a very basicand natural property that arises most of the time when one is tryingto show that two Banach (or quasi-Banach) spaces are isomorphic.However, its practical implementation depends on knowing a priorilarge classes of spaces when the property holds. And this might be anintractable problem in almost any general setting.W´ojtowicz [35] and Wojtaszczyk [34] discovered independently, witha lapse of 11 years, the following beautiful criterion in the spirit of theSchr¨oder-Bernstein problem to check whether two unconditional bases(in possibly different quasi-Banach spaces) are permutatively equiva-lent. Theorem 1.1 (see [34, Proposition 2.11] and [35, Corollary 1]) . Let ( x n ) ∞ n =1 and ( y n ) ∞ n =1 be two unconditional bases of quasi-Banach spaces X and Y . Suppose that ( x n ) ∞ n =1 is permutatively equivalent to a subba-sis of ( y n ) ∞ n =1 and that ( y n ) ∞ n =1 permutatively is equivalent to a subbasisof ( x n ) ∞ n =1 . Then ( x n ) ∞ n =1 and ( y n ) ∞ n =1 are permutatively equivalent. Inparticular, X ≈ Y . The validity of the Schr¨oder-Bernstein principle for unconditionalbases has a played a crucial role in the development of the subjectof uniqueness of unconditional basis in quasi-Banach spaces (see, e.g.,[5–9]). Casazza and Kalton brought this principle to the reader’s aware-ness in [13] and used it to give new examples of Banach spaces with aunique unconditional basis up to permutation. The simplifying powerof the Schr¨oder-Bernstein principle for unconditional bases would havemade life much easier also for all the authors who had previously workedon the problem of uniqueness of unconditional bases up to permutationand who, in order to obtain the same conclusions, had to impose ad-ditional properties to the bases in relation to other general techniquessuch as the decomposition method (see e.g. [10, Proposition 7.7]). It isindeed remarkable that, although the combinatorial arguments used byWojtaszczyk to prove Theorem 1.1 are somewhat standard, they wentunnoticed until close to the 21st century!The state of art of the Schr¨oder-Bernstein problem for Banach spacesin the pre-Gowers era was described by Casazza in [12]. His paper with
ERMUTATIVE EQUIVALENCE OF SQUARES OF UNCONDITIONAL BASES 3
Kalton [13] appeared just one year after Gowers and Maurey disprovedthe Schr¨oder-Bernstein problem for Banach spaces and Wojtaszczyk’sreinterpreted the Schr¨oder-Bernstein principle for unconditional bases.Thus, it is not surprising that the following question was timely raisedin [13]:
Question . (See [13, Remarks following the proof of Theorem 5.7])Suppose that ( x n ) ∞ n =1 and ( y n ) ∞ n =1 are two unconditional bases whosesquares are permutatively equivalent. Does it follow that ( x n ) ∞ n =1 and( y n ) ∞ n =1 are permutatively equivalent?This problem was a driving force for the present investigation andwe solve it in the affirmative. In fact we show that the result stillholds replacing the assumption on the square of the bases with theweaker assumption that some powers of the bases are permutativelyequivalent. We will do that in Section 2.Answering Question 1.2 in the positive offers a new paradigm totackle the problem of uniqueness of unconditional basis up to permu-tation in the general setting of quasi-Banach spaces. The necessary in-gredients and preparatory results leading to the main theoretical tool,namely Theorem 3.9, are presented in a self-contained fashion in Sec-tion 3.In Sections 4 and 5 we embark on a comprehensive survey of quasi-Banach spaces with a unique unconditional basis up to permutationwhich are susceptible to be applied the scheme of Section 3.In Section 6 we further exploit the usefulness of Theorem 3.9 to showthat the property of uniqueness of unconditional bases is preservedwhen we take finite direct sums of a wide class of quasi-Banach spaceswith this property. When combined with the spaces from Sections 4and 5 we obtain a myriad of new examples of spaces with uniquenessof unconditional basis up to permutation.We use standard terminology and notation in Banach space theoryas can be found, e.g., in [4]. Most of our results, however, will beestablished in the general setting of quasi-Banach spaces; the unfamiliarreader will find general information about quasi-Banach spaces in [24].We next gather the notation that it is more heavily used. In keepingwith current usage we will write c ( J ) for the set of all ( a j ) j ∈ J ∈ F J such that |{ j ∈ J : a j = 0 }| < ∞ , where F could be the real or complexscalar field. Given s ∈ N we put N [ s ] = { , . . . , s } . Given a quasi-Banach space X and s ∈ N we denote by κ [ s, X ] the smallest constant C such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) s X j =1 f j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C Ñ s X j =1 k f j k é , f j ∈ X. F. ALBIAC AND J. L. ANSORENA
The closed linear span of a subset V of X will be denoted by [ V ]. Acountable family B = ( x n ) n ∈N in X is an unconditional basic sequence if for every f ∈ [ x n : n ∈ N ] there is a unique family ( a n ) n ∈N in F suchthat the series P n ∈N a n x n converges unconditionally to f . If B is anunconditional basic sequence, there is a constant K ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X n ∈N a n x n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ K (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X n ∈N b n x n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) for any finitely non-zero sequence of scalars ( a n ) n ∈N with | a n | ≤ | b n | for all n ∈ N (see [2, Theorem 1.10]). If this inequality is satisfied fora given K we say that B is K -unconditional. If we additionally have[ x n : n ∈ N ] = X then B is an unconditional basis of X . If B is anunconditional basis of X , then the map F : X → F N , f = X n ∈N a n x n ( x ∗ n ( f )) n ∈N = ( a n ) n ∈N will be called the coefficient transform with respect to B , and the func-tionals ( x ∗ n ) n ∈N the coordinate functionals of B .Given a countable set N , we write E N := ( e n ) n ∈N for the canonicalunit vector system of F N , i.e., e n = ( δ n,m ) m ∈N for each n ∈ N , where δ n,m = 1 if n = m and δ n,m = 0 otherwise. A sequence space willbe a quasi-Banach space X ⊆ F N for which E N is a normalized 1-unconditional basis.The Banach envelope of a quasi-Banach space X consists of a Banachspace c X together with a linear contraction J X : X → c X satisfying thefollowing universal property: for every Banach space Y and every linearcontraction T : X → Y there is a unique linear contraction “ T : c X → Y such that “ T ◦ J X = T . We say that a Banach space Y is the Banachenvelope of X via the map J : X → Y if the associated map b J : c X → Y is an isomorphism.Other more specific terminology will be introduced in context whenneeded.2. Permutative equivalence of powers of unconditionalbases
Suppose that B x = ( x n ) n ∈N and B u = ( u n ) n ∈N are (countable) familiesof vectors in quasi-Banach spaces X , Y , respectively. We say that B x = ( x n ) n ∈N C - dominates B u = ( u n ) n ∈N if there is a linear map T from the closed subspace of X spanned by B x into Y with T ( x n ) = u n for all n ∈ N such that k T k ≤ C . If T is an isomorphic embedding, B x and B u are said to be equivalent . We say that B x is permutativelyequivalent to a family B y = ( y m ) n ∈M in Y , and we write B x ∼ B y , ERMUTATIVE EQUIVALENCE OF SQUARES OF UNCONDITIONAL BASES 5 if there is a bijection π : N → M such that B x and ( y π ( n ) ) n ∈N areequivalent. A subbasis of an unconditional basis B x = ( x n ) n ∈N is afamily ( x n ) n ∈M for some subset M of N .Let ( X i ) i ∈ F be a finite collection of (possibly repeated) quasi-Banachspaces. The Cartesian product L i ∈ F X i equipped with the quasi-norm k ( x i ) i ∈ F k = sup i ∈ F k x i k , x i ∈ X i is a quasi-Banach space. Suppose that B i = ( x i,n ) j ∈N i is an uncondi-tional basis of X i for each i ∈ F . Set N = [ i ∈ F { i } × N i . (2.1)Then the countable sequence L i ∈ F B i := ( x i,n ) ( i,n ) ∈N given by x i,n =( x i,n,j ) j ∈ F , where x i,n,j = x i,n if i = j, L i ∈ F X i . If F = N [ s ] and X i = X for all i ∈ F , the resulting direct sum is called the s -fold product of X and wesimply write X s = L i ∈ F X i . Similarly, if B i = B for all i ∈ F = N [ s ],we put B s = L i ∈ F B i and say that B s is the s -fold product of B . Wewill refer to the 2-fold product of a basis as to the square of that basis.We start with an elementary lemma. Lemma 2.1.
Let B = ( x n ) n ∈N be an unconditional basis of a quasi-Banach space X . For a given s ∈ N , consider the s -fold product B s = ( x i,n ) ( i,n ) ∈ N [ s ] ×N . Then, for any function α : N → N [ s ] , the basicsequence ( x α ( n ) ,n ) n ∈N (which is permutatively equivalent to a subbasisof B s ) is equivalent to B .Proof. Suppose that B is K -unconditional. If we put N i = α − ( i ) for i ∈ N [ s ] then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X n ∈N a n x α ( n ) ,n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = sup i ∈ N [ s ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X n ∈N i a n x n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , for all ( a n ) ∞ n =1 ∈ c . Hence,1 κ [ s, X ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X n ∈N a n x n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X n ∈N a n x α ( n ) ,n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ K (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X n ∈N a n x n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . (cid:3) The following version of the Hall-K¨onig Lemma (also known as
Mar-riage Lemma ) for infinite families of finite sets is essential in the proofof Theorem 2.4.
F. ALBIAC AND J. L. ANSORENA
Theorem 2.2 (see [19, Theorem 1]) . Let N be a set and ( N i ) i ∈ I be afamily of finite subsets of N . Suppose that | F | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ i ∈ F N i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for every F ⊆ I finite. Then there is a one-to-one map φ : I → N with φ ( i ) ∈ N i for every i ∈ I . Theorem 2.3.
Let B x and B y be two unconditional bases of quasi-Banach spaces X and Y respectively. Suppose that B sx is permutativelyequivalent to a subbasis of B sy for some s ≥ . Then B x is permutativelyequivalent to a subbasis of B y .Proof. Put B x = ( x n ) n ∈N , B y = ( y n ) n ∈M , B sx = ( x i,n ) ( i,n ) ∈ N [ s ] ×N and B sy = ( y i,n ) ( i,n ) ∈ N [ s ] ×M . By hypothesis there is a one-to-one map π = ( π , π ) : N [ s ] × N → N [ s ] × M such that the unconditional bases B sx and ( y π ( i,n ) ) ( i,n ) ∈ N [ s ] ×N are equiv-alent. For n ∈ N set M n = { π ( i, n ) : i ∈ N [ s ] } . If F is a finite subsetof N we have π ( N [ s ] × F ) ⊆ N [ s ] × [ n ∈ F M n , and since π is one-to-one, s | F | ≤ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ n ∈ F M n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Hence, | F | ≤ | ∪ n ∈ F M n | . We also have |M n | ≤ s for all n ∈ N .Therefore, by Theorem 2.2, there exist a one-to-one map φ : N → M ,a map α : N → N [ s ], and a map β : M → N [ s ] such that π ( α ( n ) , n ) = ( β ( n ) , φ ( n )) , n ∈ N , from where it follows that the unconditional basic sequences B ′ x =( x α ( n ) ,n ) n ∈N and B ′ y = ( y β ( n ) ,φ ( n ) ) n ∈N are equivalent. Since, on theother hand, by Lemma 2.1, B ′ x is equivalent to B and B ′ y is permuta-tively equivalent to ( y m ) m ∈M ′ , where M ′ = φ ( N ), we are done. (cid:3) Theorem 2.4.
Let B x and B y be two unconditional bases of quasi-Banach spaces X and Y . Suppose that B sx ∼ B sy for some s ≥ . Then B x ∼ B y .Proof. Apylying Theorem 2.3 yields that B x is permutatively equivalentto a subbabis of B y , and switching the roles of the basis also the otherway around. Using Theorem 1.1 closes the proof. (cid:3) ERMUTATIVE EQUIVALENCE OF SQUARES OF UNCONDITIONAL BASES 7
Corollary 2.5.
Let B be an uconditional basis of a quasi-Banach space.Suppose that B t is permutatively equivalent to a subbasis of B s for some t > s ≥ . Then B ∼ B .Proof. Since t ≥ s +1, B s +1 is permutatively equivalent to a subbasis of B s . By induction we deduce that B u +1 is permutatively equivalent toa subbasis of B u for every u ≥ s , and so by transitivity, B u is permuta-tively equivalent to a subbasis of B s for every u ≥ s . In particular, B s is is permutatively equivalent to a subbasis of B s . Therefore, by Theo-rem 2.3, B is is permutatively equivalent to a subbasis of B . Since B is permutatively equivalent to a subbasis of B , applying Theorem 1.1we are done. (cid:3) A new theoretical approach to the uniqueness ofunconditional basis in quasi-Banach spaces
From a structural point of view, it is useful to know if a given spacehas an unconditional basis and, if the answer is yes, whether this is theunique unconditional basis of the space. Recall that a quasi-Banachspace X with an unconditional basis B is said to have a unique un-conditional basis , if every semi-normalized unconditional basis of X is equivalent to B . For convenience, from now on all bases will beassumed to be semi-normalized. Note that, if B = ( x n ) n ∈N is a semi-normalized unconditional basis then it is equivalent to the normalizedbasis ( x n / k x n k ) n ∈N .For a Banach space with a symmetric basis it is rather unusual tohave a unique unconditional basis. It is well-known that ℓ has a uniqueunconditional basis [25], and a classical result of Lindenstrauss andPe lczy´nski [27] asserts that ℓ and c also have a unique unconditionalbasis. Lindenstrauss and Zippin [28] completed the picture by showingthat those three are the only Banach spaces in which all unconditionalbases are equivalent.Once we have determined that a Banach space does not have a sym-metric basis (a task that can be far from trivial) we must rethink theproblem of uniqueness of unconditional basis. In fact, an uncondi-tional non-symmetric basis admits a continuum of nonequivalent per-mutations (cf. [20, Theorem 2.1]). Hence for Banach spaces withoutsymmetric bases it is more natural to consider instead the question ofuniqueness of unconditional bases up to (equivalence and) a permuta-tion, (UTAP) for short. We say that X has a (UTAP) unconditionalbasis B if every unconditional basis in X is permutatively equivalent to B . The first movers in this direction were Edelstein and Wojtaszczyk,who proved that finite direct sums of c , ℓ and ℓ have a (UTAP) F. ALBIAC AND J. L. ANSORENA unconditional basis [16]. Bourgain et al. embarked on a comprehensivestudy aimed at classifying those Banach spaces with unique uncon-ditional basis up to permutation, that culminated in 1985 with their
Memoir [10]. They showed that the spaces c ( ℓ ), c ( ℓ ), ℓ ( c ), ℓ ( ℓ )and their complemented subspaces with unconditional basis all have a(UTAP) unconditional basis, while ℓ ( ℓ ) and ℓ ( c ) do not. However,the hopes of attaining a satisfactory classification were shattered whenthey found a nonclassical Banach space, namely the 2-convexification T (2) of Tsirelson’s space having a (UTAP) unconditional basis. Theirwork also left many open questions, most of which remain unsolved asof today.On the other hand, in the context of quasi-Banach spaces that are notBanach spaces, the uniqueness of unconditional basis seems to be thenorm rather than an exception. For instance, it was shown in [21] thata wide class of nonlocally convex Orlicz sequence spaces, including the ℓ p spaces for 0 < p <
1, have a unique uncoditional basis. The same istrue in nonlocally convex Lorentz sequence spaces ([6, 23]) and (UTAP)in the Hardy spaces H p ( T ) for 0 < p < anti-Euclidean . The subtlebut crucial role played by the lattice structure of the space in the proofof Theorem 3.9 has to be seen in that it will permit to simplify theuntangled way in which the vectors of one basis can be written interms of the other. These techniques have been extended to the non-locally convex setting and efficiently used in the literature to establishthe uniqueness of unconditional basis up to permutation of the spaces ℓ p ( ℓ q ) = ( ℓ q ⊕ ℓ p ⊕· · ·⊕ ℓ p . . . ) q for p ∈ (0 , ∪{∞} and q ∈ (0 , ∪{ , ∞} (see [5–9]), with the convention that ℓ ∞ here means c .Before moving on, recall that an unconditional basic sequence B u =( u m ) m ∈M in a quasi-Banach space X is said to be complemented if itsclosed linear span U = [ B u ] is a complemented subspace of X , i.e.,there is a bounded linear map P : X → U with P | U = Id U . Noticethe unconditional basic sequence B u = ( u m ) m ∈M is complemented in X if and only if there exists a sequence ( u ∗ m ) m ∈M in X ∗ such that u ∗ m ( u n ) = δ m,n for every ( m, n ) ∈ M and a there is a linear bounded ERMUTATIVE EQUIVALENCE OF SQUARES OF UNCONDITIONAL BASES 9 map P u : X → X given by P u ( f ) = X m ∈M u ∗ m ( f ) u m , f ∈ X. (3.1)We will refer to ( u ∗ m ) m ∈M as a sequence of projecting functionals for B u . A family B u = ( u m ) m ∈M in X with mutually disjoint supportswith respect to a given unconditional basis B is an unconditional basicsequence. In the case when, moreover, supp( u m ) is finite for every m ∈ M we say that B u is a block basic sequence (with respect to B ).We say that the block basic sequence B u is well complemented (withrespect to B ) if we can choose a sequence of projecting functionals B ∗ u = ( u ∗ m ) m ∈M with supp( u ∗ m ) ⊆ supp( u m ) for all m ∈ M . In thiscase, B ∗ u is called a sequence of good projecting functionals for B u .The following definition identifies and gives relief to an unstatedfeature shared by some unconditional bases. Examples of such basescan be found, e.g., in [6, 13, 21], where the property naturally arises inconnection with the problem of uniqueness of unconditional basis. Definition 3.1.
An unconditional basis B = ( x n ) n ∈N of a quasi-Banach space will be said to be universal for well complemented blockbasic sequences if for every semi-normalized well complemented blockbasic sequence B u = ( u m ) m ∈M of B there is a map π : M → N suchthat π ( m ) ∈ supp( u n ) for every m ∈ M , and B u is equivalent to therearranged subbasis ( x π ( m ) ) m ∈M of B .The ideas in the following definition and proposition are implicit in[21]. Definition 3.2.
An unconditional basis B = ( x n ) n ∈N of a quasi-Banach space X will be said to have the peaking property if every semi-normalized well complemented block basic sequence B u = ( u m ) m ∈M with respect to B satisfiesinf m ∈M sup n ∈N | u ∗ m ( x n ) | | x ∗ n ( u m ) | > u ∗ m ) m ∈M of good projecting functionals for B u . Proposition 3.3.
Suppose B = ( x n ) n ∈N is an unconditional basis of aquasi-Banach space X . If B has the peaking property then it is universalfor well complemented block basic sequences.Proof. Let B u = ( u m ) m ∈M be a semi-normalized well complementedblock basic sequence and B ∗ u = ( u ∗ m ) m ∈M be a sequence of good pro-jecting functionals for B u such that (3.2) holds. There is π : M → N one-to-one with inf m ∈M | x ∗ π ( m ) ( u m ) | | x ∗ π ( m ) ( u m ) | > . For m ∈ M let us put λ m = x ∗ π ( m ) ( u m ) , µ m = x π ( m ) ( u ∗ m ) , and set v m = λ m x π ( m ) , v ∗ m = µ m x ∗ π ( m ) . By [1, Lemma 3.1], B v = ( v m ) m ∈M is equivalent to B u . In particular, B v is semi-normalized so that inf m λ m > m λ m < ∞ . It followsthat B v is equivalent to ( x π ( m ) ) m ∈M . (cid:3) The last ingredient in the deconstruction process we are carrying outis the following feature about the lattice structure of a quasi-Banachspace.
Definition 3.4.
A quasi-Banach space (respectively, a quasi-Banachlattice) X is said to be sufficiently Euclidean if ℓ is crudely finitelyrepresentable in X as a complemented subspace (respectively, com-plemented sublattice), i.e., there is a positive constant C such thatfor every n ∈ N there are bounded linear maps (respectively, latticehomomorphisms) I n : ℓ n → X and P n : X → ℓ n with P n ◦ I n = Id ℓ n and k I n k k P n k ≤ C . We say that X is anti-Euclidean (resp. latticeanti-Euclidean ) if it is not sufficiently Euclidean.Any (semi-normalized) unconditional basis of a quasi-Banach space X is equivalent to the unit vector system of a sequence space andso it induces a lattice structure on X . In general, we will say thatan unconditional basis has a property about lattices if its associatedsequence space has it. And the other way around, i.e., we will say thata sequence space enjoys a certain property relevant to bases if its unitvector system does.A quasi-Banach lattice X is said to be L-convex if there is ε > f and ( f i ) ki =1 in X satisfy 0 ≤ f i ≤ f for every i = 1, . . . , k , and (1 − ε ) kf ≥ P ki =1 f i we have ε k f k ≤ max ≤ i ≤ k k f i k .Kalton [22] showed that a quasi-Banach lattice is L -convex if and onlyif it is p -convex for some p >
0. So, most quasi-Banach lattices (andunconditional bases) ocurring naturally in analysis are L-convex.The space ℓ is the simplest and most important example of anti-Euclidean space (see e.g. [1, Comments previous to Remark 2.9]). So,it is helpful to be able to count on conditions that guarantee that theBanach envelope of a given quasi-Banach space is ℓ . Lemma 3.5 (see [1, Proposition 2.10]) . Suppose X is a quasi-Banachspace with an unconditional basis B that dominates the unit vector basisof ℓ . Then the Banach envelope of X is ℓ via the coefficient transform. ERMUTATIVE EQUIVALENCE OF SQUARES OF UNCONDITIONAL BASES11
The following lemma is useful when dealing with unconditional basesthat dominate the canonical basis of ℓ .Given an unconditional basis B = ( x n ) n ∈N with coordinate function-als ( x ∗ n ) n ∈N and A ⊆ N finite we will put A [ B ] = X n ∈ A x n and ∗ A [ B ] = X n ∈ A x ∗ n . If B is clear from context we simply write A = A [ B ] and ∗ A = ∗ A [ B ]. Lemma 3.6 (cf. [5, Lemma 4.1]) . Let B = ( x n ) n ∈N be an uncondi-tional basis of a quasi-Banach space X . Suppose that B dominates thecanonical basis of ℓ . Then every semi-normalized well complementedblock basic sequence of X with respect to B is equivalent to a well com-plemented block basic sequence ( u m ) m ∈M for which ( ∗ supp( u m ) ) m ∈M isa sequence of projecting functionals.Proof. Let C be such that P n ∈N | x ∗ n ( f ) | ≤ C k f k for every f ∈ X .Set C = sup m ∈M k u m k , C = sup m ∈M k u ∗ m k , and C = sup n ∈N k x n k . Fix m ∈ M and put A m = ® n ∈ N : | u ∗ m ( x n ) | > C C ´ . We have X n ∈N \ A m | x ∗ n ( u m ) u ∗ m ( x n ) | ≤ C C X n ∈N \ A m | x ∗ n ( u m ) | ≤ . Hence, λ m : = X n ∈ A m | x ∗ n ( u m ) u ∗ m ( x n ) |≥ −
12 + X n ∈N | x ∗ n ( u m ) u ∗ m ( x n ) |≥ −
12 + u ∗ m ( u m ) = 12 . Let v m = λ − m X n ∈ A m | x ∗ n ( u m ) u ∗ m ( x n ) | x n and v ∗ m = ∗ A m . For every n ∈ N we have v ∗ m ( v m ) = 1 , λ − m | u ∗ m ( x n ) | ≤ C C , and for every n ∈ A m , 1 ≤ C C | u ∗ m ( x n ) | . Hence, the result follows from [1, Lemma 3.1]. (cid:3)
We will use the full force of the lattice structure induced by the basisin the following reduction lemma.
Lemma 3.7.
Let X be a quasi-Banach space whose Banach envelopeis anti-Euclidean. Suppose that B is an L-convex, unconditional basisof X which is universal for well complemented block basic sequences.Then, if B u is another unconditional basis of X , there are positiveintegers s and t such that B u is permutatively equivalent to a subbasisof B s and B is permutatively equivalent to a subbasis of B tu .Proof. Since B u is lattice anti-Euclidean, [5, Theorem 3.4] yields that B u is permutatively equivalent to a well complemented block basic se-quence of B s for some s ∈ N . By [1, Proposition 3.4], B s is universal forwell complemented block basic sequences so that B u is permutativelyequivalent to a subbasis of B s . Since B s inherits the convexity from B , the basis B u is L-convex and universal for well complemented blockbasic sequences. Switching the roles of B and B u yields the conclusionof the lemma. (cid:3) Remark . A remark on the inherited order structure in a quasi-Banach lattice is in order here. Kalton showed in [22, Theorem 4.2]that every unconditional basic sequence B of a quasi-Banach spacewith an L-convex unconditional basis B is L-convex. This argumentwould have, indeed, simplified the proof of Lemma 3.7. However, wewanted to make the point that the validity of the lemma does notdepend on such a deep theorem as Kalton’s.We are ready to prove the main result of this section. Theorem 3.9.
Let X be a quasi-Banach space whose Banach envelopeis anti-Euclidean. Suppose B is an unconditional basis for X such that: (i) The lattice structure induced by B in X is L-convex; (ii) B is universal for well complemented block basic sequences; and (iii) B ∼ B .Then X has a unique unconditional basis up to permutation.Proof. Let B u be another unconditional basis of X . Since B r ∼ B for every r ∈ N , applying Lemma 3.7 yields that B u is permutativelyequivalent to a subbasis of B and that B t is permutatively equivalentto a subbasis of B tu for some t ∈ N . Combining Theorem 2.3 withTheorem 1.1 yields B u ∼ B . (cid:3) Theorem 2.3 becomes instrumental in reaching the conclusion of theprevious theorem. Indeed, without it, and under the same hipotheses
ERMUTATIVE EQUIVALENCE OF SQUARES OF UNCONDITIONAL BASES13 as in Theorem 3.9, we would have only been able to guarantee thatgiven another unconditional basis B u of X , B u is permutatively equiv-alent to a subbasis of B and that B is permutatively equivalent to asubbasis of some s -fold product of B u . Thanks to Theorem 2.3 we canclose close the “gap” between B and B u and arrive at the permutativeequivalence of the two bases. Although this gap might seem small,we would like to emphasize that in the lack of Theorem 3.9 the spe-cialists were forced to use additional properties of B to infer that B isthe unique unconditional basis of X . For instance, in the proof that ℓ ( ℓ p ), 0 < p <
1, has a unique unconditional basis up to permutation,the authors used that all subbases of the canonical basis of ℓ ( ℓ p ) arepermutatively equivalent to their square (see [5]).4. Applicability of our scheme to anti-Euclidean spaces
Most anti-Euclidean spaces scattered through the literature with aunique unconditional basis (up to permutation) fulfil the hypothesesof Theorem 3.9. This can be checked on by looking up the corre-sponding references contained herein. However, with the aim to beas self-contained as possible and for the convenience of the reader wenext survey how to verify the hypotheses of Theorem 3.9 in all knownspaces (Banach and non-Banach) with a unique unconditional basisand some other new ones. The spaces in this section and the next willbe the protagonists of Section 6, where we will combine them to getthe uniqueness of unconditional basis up to permutation of their finitedirect sums.In what follows, the symbol α i . β i for i ∈ I means that the familiesof positive real numbers ( α i ) i ∈ I and ( β i ) i ∈ I verify sup i ∈ I α i /β i < ∞ . If α i . β i and β i . α i for i ∈ I we say ( α i ) i ∈ I are ( β i ) i ∈ I are equivalent,and we write α i ≈ β i for i ∈ I .4.1. The space ℓ . The simplest example of an anti-Euclidean spaceis ℓ . Since the canonical basis is perfectly homogeneous, it is univer-sal for well complemented block basic sequences. Finally, since it issymmetric, it is equivalent to its square.4.2. Orlicz sequence spaces. An Orlicz function will be a right-continuous increasing function ϕ : [0 , ∞ ) → [0 , ∞ ) such ϕ (0) = 0, ϕ (1) = 1 and ϕ ( s + t ) ≤ C ( ϕ ( s ) + ϕ ( t )) for some constant C and every s , t ≥
0. The
Orlicz space ℓ ϕ is the space associated to the Luxem-bourg quasi-norm defined from the modular ( a n ) ∞ n =1 P ∞ n =1 ϕ ( | a n | ).Our assumptions on ϕ yield that ℓ ϕ is a symmetric sequence space. Kalton proved in [21] that if ϕ satisfies t . ϕ ( t ) , ≤ t ≤ , (4.1)and Λ ϕ := lim ε → + inf
Let w = ( w n ) ∞ n =1 be a weight , i.e., asequence of positive scalars, and 0 < p < ∞ . Suppose that w decreasesto zero. The Lorentz space d ( w , p ) is the quasi-Banach space consistingof all f = ( a n ) ∞ n =1 ∈ F N such that k f k d ( w ,p ) = sup π ∈ Π ∞ X n =1 | a π ( n ) | p w n ! /p < ∞ , where Π is the set of all permutations of N . The unit vector systemis a symmetric basis of d ( w , p ). It was proved in [6] that if the weight ERMUTATIVE EQUIVALENCE OF SQUARES OF UNCONDITIONAL BASES15 fulfils the condition inf k ∈ N k X n =1 w n k p > , (4.3)then d ( w , p ) has a unique unconditional basis up to permutation. Next,we deduce this result by combining Theorem 3.9 with arguments from[6]. Proposition 4.2 (cf. [6]) . Let < p < and w = ( w n ) ∞ n =1 decreasingto zero. Then d ( w , p ) ⊆ ℓ if and only if (4.3) holds. Moreover, if (4.3) holds, then (i) the Banach envelope of d ( w , p ) is ℓ via the inclusion map, and (ii) the unit vector system of d ( w , p ) has the peaking property.Proof. For k ∈ N write s k = P kn =1 w n . Assume that d ( w , p ) ⊆ ℓ andlet C be the norm of the inclusion map. If | A | = k we have k A k = k, and k A k w ,p = s /pk . Thus k ≤ Cs /pk for every k ∈ N .The weak-Lorentz space d ∞ ( u , p ) associated to a weight u = ( u n ) ∞ n =1 and 0 < p < ∞ consists of all sequences f ∈ c whose non-increasingrearrangement ( a ∗ k ) ∞ k =1 satisfies k f k d ∞ ( u ,p ) = sup k k X n =1 u n ! /p a ∗ k < ∞ . We have d ∞ ( u , p ) ⊆ d ( u , p ) for every 0 < p < ∞ and every weight u .If u p = ( n p − ( n − p ) ∞ j =1 the rearrangement inequality and the meredefinition of the spaces yields[ d ( u p , p )] p · [ d ∞ ( u p , p )] − p ⊆ ℓ . We also have the obvious inclusion d ( u p , p ) ⊆ [ d ( u p , p )] p · [ d ( u p , p )] − p . Summing up, we obtain d ( u p , p ) ⊆ ℓ .Assume that w fulfils (4.3). We deduce that d ( w , p ) ⊆ d ( u p , p ).Therefore, d ( w , p ) ⊆ ℓ . Then, (i) follows from Lemma 3.5. To prove(ii), we pick a semi-normalized well complemented block basic sequence( u m ) m ∈M with good projecting functionals ( u ∗ m ) m ∈M . By Lemma 3.6,we can suppose that u ∗ m = ∗ supp( u m ) so thatsup n ∈ N | u ∗ m ( e n ) | | e ∗ n ( u m ) | = sup n ∈ N | e ∗ n ( u m ) | . Finally, note that the proof of [6, Theorem 2.4] givesinf m ∈M sup n ∈ N | e ∗ n ( u m ) | > . (cid:3) Tsirelson’s space.
Casazza and Kalton established in [13] theuniqueness of unconditional basis up to permutation of Tsirelson’sspace T and its complemented subspaces with unconditional basis asa byproduct of their study of complemented basic sequences in latticeanti-Euclidean Banach spaces. Their result answered a question byBourgain et al. ([10]), who had proved the uniqueness of unconditionalbasis up to permutation of the 2-convexifyed Tsirelson’s space T (2) of T (see Example 5.10 in § T (2) , which is“highly” Euclidean, the space T is anti-Euclidean. To see the latterrequires the notion of dominance, introduced in [13].Let B = ( x n ) ∞ n =1 be a (semi-normalized) unconditional basis of aquasi-Banach space X . Given f , g ∈ X , we write f ≺ g if m < n forall m ∈ supp( f ) and n ∈ supp( g ). The basis B is said to be left (resp.right) dominant if there is a constant C such that whenever ( f i ) Ni =1 and( g i ) Ni =1 are disjointly supported families with f i ≺ g i (resp. g i ≺ f i )and k f i k ≤ k g i k for all i ∈ N [ N ], then k P Ni =1 f i k ≤ C k P Ni =1 g i k . If X is a Banach space with a left (resp. right) dominant unconditionalbasis B there is a unique r = r ( B ) ∈ [1 , ∞ ] such that ℓ r is finitelyblock representable in X . In the case when r ( B ) ∈ { , ∞} , X is anti-Euclidean (see [13, Proposition 5.3]).The canonical basis of the Tsirelson space T is right dominant [13,Proposition 5.12], and r ( T ) = 1. Moreover, by [13, Proposition 5.5]and [15, page 14], the canonical basis (as well as each of its subases)is equivalent to its square. In our language, [13, Theorem 5.6] saysthat every left (resp. right) dominant unconditional basis is universalfor well complemented block basic sequences. Finally, since it is locallyconvex, T is trivially an L-convex lattice.4.5. Bourgin-Nakano spaces.
Let N be a countable set. A Bourgin-Nakano index is a family ( p n ) n ∈N in (0 , ∞ ) with p = inf n p n >
0. The
Bourgin-Nakano space ℓ ( p n ) is the quasi-Banach space built from themodular m ( p n ) : F N → [0 , ∞ ) , ( a n ) n ∈N X n ∈N | a n | p n . Note that, by the Monotone Convergence Theorem, the closed unit ballof ℓ ( p n ) is the set B ℓ ( p n ) = { f ∈ F N : m ( p n ) ( f ) ≤ } . ERMUTATIVE EQUIVALENCE OF SQUARES OF UNCONDITIONAL BASES17
If we endow ℓ ( p n ) with the natural ordering, it becomes a p -convexquasi-Banach lattice. The separable part h ( p n ) = [ e n : n ∈ N ] of ℓ ( p n )is a sequence space. We have ℓ ( p n ) = h ( p n ) if and only if sup n p n < ∞ .These spaces where introduced by Bourgin [11] in the particular casewhen p n ≤ n ∈ N . Nakano [29] studied the case when p n ≥ n ∈ N , so that the arising spaces are locally convex,i.e., Banach spaces.Let us record some results on Bourgin-Nakano spaces of interest forthe purposes of this paper. Lemma 4.3.
Let ( p n ) n ∈N and ( q m ) m ∈M be Bourgin-Nakano indexes.Let B u = ( u j ) ∞ j =1 and B v = ( v j ) ∞ j =1 be normalized block basic sequencesin ℓ ( p n ) and ℓ ( q n ) respectively. Suppose that p n ≤ q m for all ( n, m ) ∈ supp( u j ) × supp( v j ) and all j ∈ N . Then B u -dominates B v .Proof. Let j ∈ N . Pick r j ∈ [1 , ∞ ) such that p n ≤ r ≤ q m for every n ∈ A j := supp( u j ) and m ∈ B j := supp( v j ). Put u j = P n ∈ A j a j e j and v j = P n ∈ A j b j e j . Since k u j k = k v j k = 1, we have X n ∈ A j | a j | p n = 1 = X m ∈ B j | b m | q m . Let f = P ∞ j =1 c j u j ∈ B ℓ ( p n ) . We have | c j | ≤ j ∈ N .Hence, m ( q n ) Ñ ∞ X j =1 c j v j é = ∞ X j =1 X m ∈ B j | c j | q m | b m | q m ≤ ∞ X j =1 | c j | r X m ∈ B j | b m | q m = ∞ X j =1 | c j | r X n ∈ A j | a n | p n ≤ ∞ X j =1 X n ∈ A j | c j | p n | a n | p n ≤ . Therefore, P ∞ j =1 c j u j ∈ B ℓ ( q n ) . (cid:3) Proposition 4.4 (see [13, Proof of Theorem 5.8]) . Let ( p n ) ∞ n =1 be anon-increasing (resp. non-decreasing) Bourgin-Nakano index. Then,the unit vector system of ℓ ( p n ) is right (resp. left) dominant. Moreover, r ( ℓ ( p n )) = lim n p n .Proof. It is a consequence of Lemma 4.3. (cid:3)
Given ( p n ) n ∈N we put ( c p n ) n ∈N = (max { , p n } ) n ∈N . Proposition 4.5.
Let ( p n ) n ∈N be a Bourgin-Nakano index. Then theBanach envelope of ℓ ( p n ) is ℓ ( c p n ) via the inclusion map.Proof. Put N b = { n ∈ N : p n < } , N k = { n ∈ N : p n ≥ } . Theobvious map from F N onto F N b × F N k restricts to a lattice isomorphismfrom ℓ ( p n ) onto ℓ ( p n ) n ∈N b ⊕ ℓ ( p n ) n ∈N k . Hence, by [1, Lemma 2.3], wecan assume without loss of generality that N k = ∅ . In this particularcase, since P n ∈N | a n | ≤ a n ) n ∈N ∈ B ℓ ( p n ) and e n ∈ B ℓ ( p n ) for every n ∈ N , the closed convex hull of B ℓ ( p n ) in ℓ ( N ) is the unitclosed ball of ℓ ( N ). Since ℓ ( c p n ) = ℓ ( N ) isometrically, we infer thatthe Banach envelope of ℓ ( p n ) is ℓ ( c p n ) isometrically under the inclusionmap. (cid:3) Corollary 4.6.
Let ( p n ) n ∈N be a Bourgin-Nakano index. Suppose that lim sup n p n ≤ . Then, the Banach envelope of ℓ ( p n ) is anti-Euclidean.Proof. Just combine Propositions 4.4 and 4.5. (cid:3)
Proposition 4.7.
Let ( p n ) ∞ n =1 be a Bourgin-Nakano index. Then, theunit vector system of ℓ ( p n ) is universal for well complemented blockbasic sequences.Proof. Let B y = ( y m ) m ∈M be a semi-normalized well complementedblock basic sequence and let ( u ∗ m ) m ∈M be a sequence of good projectingfunctionals. Since X n ∈N e ∗ n ( y m ) y ∗ m ( e n ) = y ∗ m ( y m ) = 1for every m ∈ M , there are families ( A m ) m ∈M and ( B m ) m ∈M of subsetsof N and π : M → N such that, if λ m = X n ∈ A m e ∗ n ( y m ) y ∗ m ( e n ) and µ m = X n ∈ B m e ∗ n ( y m ) y ∗ m ( e n ) , then min {| λ m | , | µ m |} ≥ / A m ∪ B m = supp( y m ), A m ∩ B m = { π ( m ) } ,and max n ∈ A m p n = min n ∈ B m p n = p π ( m ) for every m ∈ M . Let u m = S A m ( y m ), u ∗ m = S ∗ A m ( y ∗ m ), v m = S B m ( y m ), and v ∗ m = S ∗ B m ( y m ) for m ∈ M . Since u ∗ m ( u m ) = λ m and v ∗ m ( v m ) = µ m for every m ∈ M , applying [1, Lemma 3.1] yieldsthat both B u = ( u m ) m ∈M and B v = ( v m ) m ∈M are well complementedblock basic sequences equivalent to B y . By Lemma 4.3, B u dominates B := ( e π ( m ) ) m ∈M and, in turn, B dominates B v . We infer that B y and B are equivalent. (cid:3) ERMUTATIVE EQUIVALENCE OF SQUARES OF UNCONDITIONAL BASES19
Proposition 4.8.
Let ( p n ) n ∈N be a Bourgin-Nakano index. The unitvector system of ℓ ( p n ) is equivalent to its square if and only if there isa partition ( N , N ) of N and bijections π i : N → N i , i = 1 , , suchthat, for some < c < , X n ∈N c pnqi,n | pn − qi,n | < ∞ , i = 1 , , where if q i,n = p π i ( n ) .Proof. This result follows from [30, Theorem 1], which characterizeswhen two (a priori different) Bourgin-Nakano spaces are identical. (cid:3)
We remark that, in certain cases, we can give a more simple char-acterization of Nakano spaces that are lattice isomorphic to its square.For instance, if ( p n ) ∞ n =1 is a monotone sequence, then ℓ ( p n ) is latticeisomorphic to its square if and only if (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p n − p n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .
11 + log( n ) , n ∈ N (see [13, Proof of Theorem 5.8]). Theorem 4.9.
Suppose that Bourgin-Nakano index ( p n ) n ∈N satisfies lim sup n p n ≤ . Suppose also that there exist a partition ( N , N ) of N , and bijections π i : N → N i , i = 1 , , so that X n ∈N c / | p n − p πi ( n ) | < ∞ , i = 1 , , for some < c < . Then ℓ ( p n ) has a unique unconditional basis up topermutation.Proof. Just combine Corollary 4.6, Proposition 4.7, Proposition 4.8 andTheorem 3.9. (cid:3)
An important class of anti-Euclidean spaces arises from a specialtype of bases called strongly absolute. We tackle this case separatelyin the next section.5.
Applicability to spaces with strongly absolute bases
In the category of bases one could say that strongly absolute bases are“purely nonlocally convex” bases, in the sense that if a quasi-Banachspace X has a strongly absolute basis, then its unit ball is far frombeing a convex set and so X is far from being a Banach space. Theterm strongly absolute for a basis was coined in [23]. Here we givea slightly different, but equivalent, definition. We say that a (semi-normalized) unconditional basis B = ( x n ) n ∈N of a quasi-Banach space X is strongly absolute if for every ε > < C ( ε )such that X n ∈N | x ∗ n ( f ) | ≤ max ® C ( ε ) sup n ∈N | x ∗ n ( f ) | , ε k f k ´ , f ∈ X. (5.1)In the following lemma we record a key property of strongly absolutebases. The proof is straightforward and so we omit it. Lemma 5.1.
Let B = ( x n ) n ∈N be a strongly absolute unconditionalbasis of a quasi-Banach space X . Suppose that V ⊆ X is such that inf f ∈ V k f k − kF ( f ) k > . Then, inf f ∈ V k f k − kF ( f ) k ∞ > . Proposition 5.2 (cf. [23]) . Let B be a strongly absolute unconditionalbasis of a quasi-Banach space X . Then: (i) The Banach envelope of X is ℓ via the coefficient transform. (ii) B has the peaking property.Proof. It is clear that B dominates the unit vector system of ℓ , so that(i) follows from Lemma 3.5.Let B u = ( u m ) m ∈M be a semi-normalized well complemented blockbasic sequence. By Lemma 3.6 we may assume that ( u ∗ m ) m ∈M =( ∗ supp( u m ) ) m ∈M is a sequence of good projecting functionals for B u .Using (i) and [1, Lemma 2.1] we deduce that the sequence ( F ( u m )) ∞ m =1 is semi-normalized in ℓ . Therefore,inf m k u m k − kF ( u m ) k > . Lemma 5.1 yieldsinf m ∈M sup n ∈N | u ∗ m ( x n ) | | x ∗ n ( u m ) | = inf m ∈M kF ( u m ) k ∞ ≥ inf m ∈M k u m k inf m ∈M kF ( u m ) k ∞ k u m k > . (cid:3) Combining Proposition 5.2 with Theorem 3.9 immediately yields thefollowing general result.
Corollary 5.3.
Let X be a quasi-Banach space with a strongly absoluteunconditional basis which induces an L-convex structure on X . If B isequivalent to its square, then X has a unique unconditional basis up topermutation. Wojtaszczyk obtained in [34] the uniqueness of unconditional basis ofa quasi-Banach space X under the same hypotheses as in Corollary 5.3replacing B ∼ B with the weaker assumption that X s ≈ X for some s ≥
2. For the sake of completeness, we next show how we can combinethe techniques from [34] to pass from the condition “ X s ≈ X for some s ≥
2” to “ B ∼ B ”. ERMUTATIVE EQUIVALENCE OF SQUARES OF UNCONDITIONAL BASES21
Theorem 5.4 (cf. [34, Theorem 2.12]) . Let X be a quasi-Banach spacewith a strongly absolute unconditional basis B that induces an L -convexlattice structure on X . If X s ≈ X for some s ≥ then B ∼ B ; inparticular X ≈ X .Proof. Put B s = ( y m ) m ∈M . We have that B s = ( y i,m ) ( i,m ) ∈ N [ s ] ×M is permutatively equivalent to a basis of X ≈ X s . Hence, by [34,Proposition 2.10], there is α : M → N [ s ] such that B ′ = ( y α ( m ) ,m ) m ∈M is permutatively equivalent to a subbasis of B . By Lemma 2.1, B s isequivalent to B ′ . Since B is permutatively equivalent to a subbasis of B and B is permutatively equivalent to a subbasis of B s , applyingTheorem 1.1 yields B s ∼ B ∼ B . (cid:3) As we said before, a strongly absolute unconditional basis can bethought of as a basis that dominates the canonical basis of ℓ but it isfar from it. This intuition is substantiated by the following elementaryresult whose proof we omit. Lemma 5.5.
Let B x and B y be unconditional bases of quasi-Banachspaces X and Y respectively. Suppose that B x dominates B y and that B y is strongly absolute. Then B y is strongly absolute. To complement the theoretical contents of this section we shall in-troduce a quantitative tool from approximation theory that measureshow far an unconditional basis is from the canonical ℓ -basis.Given an unconditional basis B of a quasi-Banach space X , its lowerdemocracy function is defined as ϕ lm [ B ] = inf | A |≥ m k A [ B ] k , m ∈ N . Note that if B is strongly absolute thenlim m →∞ m ϕ lm [ B ] = ∞ . The following result establishes that, conversely, if ( ϕ lm [ B ]) ∞ m =1 is suffi-ciently far away from the sequence ( m ) ∞ m =1 , then the basis B is stronglyabsolute. Proposition 5.6.
Let B = ( x n ) ∞ n =1 be an unconditional basis of aquasi-Banach space X . Suppose that there exists < p < such thatfor some constant < C we have m /p ≤ Cϕ lm [ B ] , m ∈ N . Then B is strongly absolute. Proof.
We may regard X as a sequence space whose basis B is just theunit vector system. Pick r ∈ ( p, X ⊆ ℓ p, ∞ ⊆ ℓ r continuously. Since the canonical basis of ℓ r is strongly absolute (see[26, Lemma 2.2]), by Lemma 5.5 the proof is over. (cid:3) We will use Proposition 5.6 to readily deduce that the followingimportant examples of bases, which are permutatively equivalent totheir square, are strongly absolute.
Example . Given 0 < p i < i ∈ N [ n ], the canonical basis of themixed norm space ℓ p ( · · · ℓ p i ( · · · ( ℓ p n ))) is unconditional, strongly abso-lute, and induces a structure of L-convex lattice on the whole space. Example . Let d ∈ N . The canonical basis B of the Hardy spaces H p ( T d ), 0 < p < m /p ≈ ϕ lm [ B , H p ( T d )] , m ∈ N . Hence, B is strongly absolute. Example . Given a dimension d ∈ N , let Θ d = { , } d \ { } andconsider the set of indicesΛ d = Z × Z d × Θ d . The homogeneous Triebel-Lizorkin sequence space ˚ t s,dp,q of indeces and p ∈ (0 , ∞ ) and q ∈ (0 , ∞ ] and smoothness s ∈ R consists of all scalarsequences f = ( a λ ) λ ∈ Λ for which k f k t sp,q = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ñ ∞ X j = −∞ X δ ∈ Θ d X n ∈ Z d jq ( s + d/ | a j,n,δ | q χ Q ( j,n ) é /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p < ∞ , were Q ( j, n ) denotes the cube of length 2 − j whose lower vertex is 2 − j n .If we restrict ourselves to non-negative “levels” j and we add ℓ p asa component we obtain the inhomegeneous Triebel-Lizorkin sequencespaces. To be precise, setΛ + d = { ( j, n, δ ) ∈ Λ d : j ≥ } , and define t s,dp,q = ℓ p ( Z d ) ⊕ { f = ( a λ ) λ ∈ Λ + d : k f k t sp,q < ∞} . It is known that the wavelet transforms associated to certain waveletbases normalized in the L -norm are isomorphisms from F sp,q ( R d ) (re-spectively ˚ F sp,q ( R d ) onto t sp,q ( R d ) (resp., ˚ t s,dp,q ). See [17, Theorem 7.20]for the homegeneous case and [33, Theorem 3.5] for the inhomogenous ERMUTATIVE EQUIVALENCE OF SQUARES OF UNCONDITIONAL BASES23 case. Thus, Triebel-Lizorkin spaces are isomorphic to the correspond-ing sequence spaces, and the aforementioned wavelet bases (regardedas distributions on Triebel-Lizorkin spaces) are equivalent to the unitvector systems of the corresponding sequence spaces.A similar technique to the one used by Temlyakov in [32] to provethat the Haar system is a democratic basis for L p when 1 < p < ∞ allows us to prove that the unique vector system E of ˚ t s,dp,q satisfies m /p ≈ ϕ lm [ E , ˚ t s,dp,q ] , m ∈ N . Consequently, if p <
1, the unique vector system of both ˚ t s,dp,q and t s,dp,q is a strongly absolute unconditional basis. Example . Given 0 < p < ∞ , the p -convexified Tsirelson’s space ,denoted T ( p ) , is obtained from T by putting k x k T ( p ) = k ( | a n | p ) ∞ n =1 k /p T (5.2)for those sequences of real numbers x = ( a n ) ∞ n =1 such that ( | a n | p ) ∞ n =1 ∈T . Equation (5.2) defines a norm for 1 ≤ p and a p -norm when 0
1. Obviously, the space ( T (1) , k · k T (1) ) is simply ( T , k · k T ).For 0 < p < ∞ , the canonical basis E of T ( p ) is 1-unconditional, itis permutatively equivalent to its square, and satisfies m /p ≈ ϕ lm [ E , T ( p ) ] , m ∈ N . Hence in particular if 0 < p < E is strongly absolute.6. Uniqueness of unconditional basis of sums ofanti-Euclidean spaces
Our last application of Theorem 3.9 establishes that the uniqueness ofunconditional bases up to permutation of anti-Euclidean quasi-Banachspaces is preserved by finite direct sums.
Theorem 6.1.
Let ( X i ) i ∈ F be a finite family of quasi-Banach spaceswhose Banach envelopes are anti-Euclidean. Suppose that for each i ∈ F , B i is an unconditional basis of X i such that (i) The lattice structure induced by B i in X i is L-convex; (ii) B i is universal for well complemented block basic sequences; and (iii) B i ∼ B i .Then the space L i ∈ F X i has a unique unconditional basis up to permu-tation.Proof. Combining [14, Proposition 2.4] and [1, Lemma 2.3] we see thatthe Banach envelope of X = L i ∈ F X i is anti-Euclidean. It is clear thatthe basis B = L i ∈ F B i is L-convex and permutatively equivalent to its square. By [1, Proposition 3.4], B is universal for well complementedblock basic sequences. So, the result follows from Theorem 3.9. (cid:3) Merging the results from Sections 4 and 5 with Theorem 6.1 providesnew additions to the list of spaces with unique unconditional basis upto a permutation.
Corollary 6.2.
Let F be a finite set of indeces. Suppose that for each i ∈ F , X i is one of the following spaces: (i) ℓ ϕ , where ϕ verifies (4.1) and (4.2) , in particular ℓ p for p ≤ ; (ii) d ( w , p ) , where w verifies (4.3) ; (iii) T ; (iv) ℓ ( p n ) , where ( p n ) ∞ n =1 verifies the hypothesis of Theorem 4.9; (v) ℓ p ( · · · ℓ p i ( · · · ( ℓ p n ))) , where < p i < for i ∈ N [ n ] ; (vi) H p ( T d ) for d ∈ N and < p < ; (vii) ˚ t s,dp,q or t s,dp,q as in Example 5.9; (viii) T ( p ) for < p < .Then X = L i ∈ F X i has a unique unconditional basis up to permutation. References [1] F. Albiac and J. L. Ansorena,
Projections and unconditional bases in directsums of ℓ p spaces, < p ≤ ∞ , arXiv:1909.06829 [math.FA].[2] F. Albiac, J. L. Ansorena, P. Bern´a, and P. Wojtaszczyk, Greedy approximationfor biorthogonal systems in quasi-Banach spaces , arXiv:1903.11651 [math.FA].[3] F. Albiac, J. L. Ansorena, S. J. Dilworth, and D. Kutzarova,
Banach spaceswith a unique greedy basis , J. Approx. Theory (2016), 80–102.[4] F. Albiac and N. J. Kalton,
Topics in Banach space theory, 2nd revised and up-dated edition , Graduate Texts in Mathematics, vol. 233, Springer InternationalPublishing, 2016.[5] F. Albiac, N. J. Kalton, and C. Ler´anoz,
Uniqueness of the unconditional basisof ℓ ( ℓ p ) and ℓ p ( ℓ ) , < p <
1, Positivity (2004), no. 4, 443–454.[6] F. Albiac and C. Ler´anoz, Uniqueness of unconditional basis in Lorentz se-quence spaces , Proc. Amer. Math. Soc. (2008), no. 5, 1643–1647.[7] ,
An alternative approach to the uniqueness of unconditional basis of ℓ p ( c ) for < p <
1, Expo. Math. (2010), no. 4, 379–384.[8] , Uniqueness of unconditional bases in nonlocally convex ℓ -products , J.Math. Anal. Appl. (2011), no. 2, 394–401.[9] , Uniqueness of unconditional bases in nonlocally convex c -products ,Israel J. Math. (2011), 79–91.[10] J. Bourgain, P. G. Casazza, J. Lindenstrauss, and L. Tzafriri, Banach spaceswith a unique unconditional basis, up to permutation , Mem. Amer. Math. Soc. (1985), no. 322, iv+111.[11] D.G. Bourgin, Linear topological spaces , Amer. J. Math. (1943), 637–659.[12] P. G. Casazza, The Schroeder-Bernstein property for Banach spaces , Banachspace theory (Iowa City, IA, 1987), Contemp. Math., vol. 85, Amer. Math.Soc., Providence, RI, 1989, pp. 61–77.
ERMUTATIVE EQUIVALENCE OF SQUARES OF UNCONDITIONAL BASES25 [13] P. G. Casazza and N. J. Kalton,
Uniqueness of unconditional bases in Banachspaces , Israel J. Math. (1998), 141–175.[14] ,
Uniqueness of unconditional bases in c -products , Studia Math. (1999), no. 3, 275–294.[15] P. G. Casazza and T. J. Shura, Tsirel ′ son’s space , Lecture Notes in Mathemat-ics, vol. 1363, Springer-Verlag, Berlin, 1989.[16] I. S. `Edel ′ ˇste˘ın and P. Wojtaszczyk, On projections and unconditional bases indirect sums of Banach spaces , Studia Math. (1976), no. 3, 263–276.[17] M. Frazier, B. Jawerth, and G. Weiss, Littlewood-Paley theory and the studyof function spaces , CBMS Regional Conference Series in Mathematics, vol. 79,Published for the Conference Board of the Mathematical Sciences, Washington,DC; by the American Mathematical Society, Providence, RI, 1991.[18] W. T. Gowers and B. Maurey,
Banach spaces with small spaces of operators ,Math. Ann. (1997), no. 4, 543–568.[19] M. Hall Jr.,
Distinct representatives of subsets , Bull. Amer. Math. Soc. (1948), 922–926.[20] J. Hennefeld, On nonequivalent normalized unconditional bases for Banachspaces , Proc. Amer. Math. Soc. (1973), 156–158.[21] N. J. Kalton, Orlicz sequence spaces without local convexity , Math. Proc. Cam-bridge Philos. Soc. (1977), no. 2, 253–277.[22] , Convexity conditions for nonlocally convex lattices , Glasgow Math. J. (1984), no. 2, 141–152.[23] N. J. Kalton, C. Ler´anoz, and P. Wojtaszczyk, Uniqueness of unconditionalbases in quasi-Banach spaces with applications to Hardy spaces , Israel J. Math. (1990), no. 3, 299–311 (1991).[24] N. J. Kalton, N. T. Peck, and J. W. Roberts, An F -space sampler , Lon-don Mathematical Society Lecture Note Series, vol. 89, Cambridge UniversityPress, Cambridge, 1984.[25] G. K¨othe and O. Toeplitz, Lineare Raume mit unendlich vielen Koordinatenund Ringen unendlicher Matrizen , J. Reine Angew Math. (1934), 193–226.[26] C. Ler´anoz,
Uniqueness of unconditional bases of c ( ℓ p ) , < p <
1, StudiaMath. (1992), no. 3, 193–207.[27] J. Lindenstrauss and A. Pe lczy´nski,
Absolutely summing operators in L p -spacesand their applications , Studia Math. (1968), 275–326.[28] J. Lindenstrauss and M. Zippin, Banach spaces with a unique unconditionalbasis , J. Functional Analysis (1969), 115–125.[29] H. Nakano, Modulared Semi-Ordered Linear Spaces , Maruzen Co., Ltd., Tokyo,1950.[30] ,
Modulared sequence spaces , Proc. Japan Acad. (1951), 508–512.[31] A. Pe lczy´nski, Universal bases , Studia Math. (1969), 247–268.[32] V. N. Temlyakov, The best m -term approximation and greedy algorithms , Adv.Comput. Math. (1998), no. 3, 249–265.[33] H. Triebel, Theory of function spaces. III , Monographs in Mathematics,vol. 100, Birkh¨auser Verlag, Basel, 2006.[34] P. Wojtaszczyk,
Uniqueness of unconditional bases in quasi-Banach spaces withapplications to Hardy spaces. II , Israel J. Math. (1997), 253–280.[35] M. W´ojtowicz, On the permutative equivalence of unconditional bases in F -spaces , Funct. Approx. Comment. Math. (1988), 51–54. Department of Mathematics, Statistics and Computer Sciences, andInaMat, Universidad P´ublica de Navarra, Pamplona 31006, Spain
E-mail address : [email protected] Department of Mathematics and Computer Sciences, Universidad deLa Rioja, Logro˜no 26004, Spain
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