Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces
aa r X i v : . [ m a t h . F A ] F e b UNIQUENESS OF UNCONDITIONAL BASIS OFINFINITE DIRECT SUMS OF QUASI-BANACHSPACES
F. ALBIAC AND J. L. ANSORENA
Abstract.
This paper is devoted to providing a unifying ap-proach to the study of the uniqueness of unconditional bases, upto equivalence and permutation, of infinite direct sums of quasi-Banach spaces. Our new approach to this type of problem permitsto show that a wide class of vector-valued sequence spaces havea unique unconditional basis up to a permutation. In particular,solving a problem from [11] we show that if X is quasi-Banach spacewith a strongly absolute unconditional basis then the infinite directsum ℓ ( X ) has a unique unconditional basis up to a permutation,even without knowing whether X has a unique unconditional basisor not. Applications to the uniqueness of unconditional structureof infinite direct sums of non-locally convex Orlicz and Lorentz se-quence spaces, among other classical spaces, are also obtained asa by-product of our work. Introduction and background
Given a Banach space (or, more generally, a quasi-Banach space) X with a normalized unconditional basis ( x n ) ∞ n =1 , let us write X ∈ U ifevery normalized unconditional basis of X is equivalent to a permuta-tion ( x π ( n ) ) ∞ n =1 of the basis ( x n ) ∞ n =1 . If we impose a stronger uniquenessproperty, where it is required that π be the identity, we write X ∈ U S .Notice that X ∈ U S if and only if X has a symmetric basis and X belongs to U . In the context of Banach spaces it is well known that X ∈ U S if and only if X is isomorphic to one of the spaces from theset S = { c , ℓ , ℓ } ([28, 29]). However, for quasi-Banach spaces which Mathematics Subject Classification.
Key words and phrases. uniqueness, unconditional basis, equivalence of bases,quasi-Banach space, Banach lattice.F. Albiac acknowledges the support of the Spanish Ministry for Science and In-novation under Grant PID2019-107701GB-I00 for
Operators, lattices, and structureof Banach spaces . F. Albiac and J. L. Ansorena acknowledge the support of theSpanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for
An´alisis Vectorial, Multilineal y Aproximaci´on . are not Banach spaces the situation is quite different since there is awide class of non-locally convex Orlicz sequence spaces, including thespaces ℓ p for 0 < p <
1, which belong to U S ([21]).Bourgain et al. studied in [13] the class Z of those Banach spaceswhich can be obtained by taking the infinite direct sum of a space from S in the sense of a space also in S , and gave a complete descriptionof the class Z ∩ U by proving that the spaces c ( ℓ ), ℓ ( c ), c ( ℓ )and ℓ ( ℓ ) belong to U , while ℓ ( ℓ ) and ℓ ( c ) do not. Many of thequestions the authors formulated in their 1985 Memoir remain openas of today. They conjectured that if a Banach space X belongs to U then so does the iterated copy of X in the sense of one of thespaces from S . This conjecture was disproved in the general casein 1999 by Casazza and Kalton, who showed that Tsirelson’s space T ∈ U whereas c ( T ) / ∈ U ([16]). Casazza and Kalton’s work gavethus continuity to a research topic that was central in Banach spacetheory in the 1960’s and 1970’s, but that was interrupted after the Memoir . Perhaps the researchers felt discouraged to put effort intoa subject that required the discovery of novel tools in order to makeheadway, with little hope for attaining a satisfactory classification ofthe Banach spaces belonging to U .At the same time, the positive results on uniqueness of unconditionalbasis obtained in the context of non-locally convex quasi-Banach spacesmotivated further study with a number of authors contributing to thedevelopment of a coherent theory. An important advance was the paper[24] by Kalton et al. followed by the work of Ler´anoz [27], who provedthat c ( ℓ p ) ∈ U for all 0 < p <
1, and Wojtaszczyk [36], who provedthat the Hardy space H p ( T ) also belongs to the class U for 0 < p < ℓ p ( ℓ ), ℓ p ( ℓ ), and ℓ ( ℓ p ) also belongto U for all 0 < p < of an atomic quasi-Banach lattice whose unit vector system is stronglyabsolute, while in Section 5 we concentrate in ℓ -sums of quasi-Banachspaces with strongly absolute bases. A brief digression could help thereader to understand better our approach in these theoretical sections.An infinite direct sum X = ( L ∞ j =1 X j ) L of quasi-Banach spaces ( X j ) ∞ j =1 in the sense of some quasi-Banach lattice L may be regarded as an in-finite matrix whose j th row is occupied by the vectors in X j . Sincethe spaces X j come with a basis X j , the vectors in X j are sequencesof scalars (relative to the basis X j ). Understanding the geometry of X often requires working simultaneously with several (or even all) rows of X and in doing so, we need to count on estimates for the bases X j ). Un-derstanding the geometry of X often requires working simultaneouslywith several (or even all) rows of X and in doing so, we need to counton estimates for the bases and spaces and spaces that do not dependon the specific row(s) we are looking at. This compels us to introducethe quantitative versions of the notions we will use and to keep track ofthe constants involved in our arguments. Finally, Section 6 is devotedto applying our theoretical schemes to practical cases. Among the vastamount of novel examples that we can tailor, we exhibit a selection ofimportant new examples of spaces that belong to U and which involveLebesgue sequence spaces, Lorentz sequence spaces, Orlicz sequencespaces, Bourgin-Nakano spaces, Hardy spaces, and Tsirelson’s space.2. Terminology
We use standard terminology and notation in Banach space theoryas can be found, e.g., in [6]. 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 [25].In keeping with 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 can be thereal or complex scalar field. The convex hull of a subset Z of a vectorspace will be denoted by co( Z ). A quasi-norm on a vector space X over F is a map k · k : X → [0 , ∞ ) satisfying k x k > f = 0, k t f k = | t | k f k for all t ∈ F and all f ∈ X , and k f + g k ≤ κ ( k f k + k g k ) , f, g ∈ X, (2.1)for some constant κ ≥
1. The optimal constant such that (2.1) holdswill be called the modulus of concavity of X . If k · k verifies k f + g k p ≤ k f k p + k g k p , f, g ∈ X, for some 0 < p ≤
1, the quasi-norm k · k is said to be a p -norm. Note that a p -norm is a quasi-norm with modulus of concavity at most F. ALBIAC AND J. L. ANSORENA /p − . If X is complete with the metric topology induced inducedby the quasi-norm, ( X, k · k ) is said to be a quasi-Banach space . A p -Banach space will be a quasi-Banach space equipped with a p -norm.The closed unit ball of a quasi-Banach space X will be denoted by B X and the closed linear span of a subset Z of X will be denoted by [ Z ].We will frequently index unconditional bases and basic sequences byan unordered countable index set N which need not be the naturalnumbers N . A countable family X = ( x n ) n ∈N in X is an unconditionalbasic sequence if for every f ∈ [ x n : n ∈ N ] there is a unique family( a n ) n ∈N in F such that the series P n ∈N a n x n converges unconditionallyto f . If X = ( x n ) n ∈N is an unconditional basic sequence, there is aconstant K ≥ (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) ≤ K (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) for all finitely non-zero sequence of scalars ( a n ) n ∈N with | a n | ≤ | b n | for all n ∈ N (see [3, Theorem 1.10]). If this condition is satisfiedsome K ≥ X is K -unconditional and if, additionally,[ x n : n ∈ N ] = X then X is said to be an unconditional basis of X . Anunconditional basis X = ( x n ) n ∈N in X becomes 1-unconditional underthe renorming k f k u = sup ((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) : | a n | ≤ | x ∗ n ( f ) | ) , f ∈ X. Thus, we will in general take the viewpoint that an unconditional basisin a quasi-Banach space X confers the structure of an atomic quasi-Banach lattice on X .If X = ( x n ) n ∈N is an unconditional basis of X with biorthogonalfunctionals ( x ∗ n ) n ∈N , the map F : X → F N given by 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 X . The support of f ∈ X with respect to X is the setsupp( f ) = { n ∈ N : x ∗ n ( f ) = 0 } , and the support of a functional f ∗ ∈ X ∗ with respect to X is the setsupp( f ∗ ) = { n ∈ N : f ∗ ( x n ) = 0 } . Given a countable set J , we write E J := ( e j ) j ∈J for the canonicalunit vector system of F J , i.e., e j = ( δ j,k ) k ∈J for each j ∈ J , where δ j,k = 1 if j = k and δ j,k = 0 otherwise. A sequence space on J will be a quasi-Banach lattice L ⊆ F J for which the 1-unconditionalbasic sequence E J is normalized. If c is dense in L , so that E J isa normalized 1-unconditional basis of L , we say that L is a minimalsequence space . The most important examples of minimal sequencespaces L on a set J are the classical Lebesgue sequence spaces ℓ p ( J )for 0 < p < ∞ , and c ( J ). As is customary, ℓ p will stand for the space ℓ p ( N ) and ℓ sp will denote ℓ p ( { n ∈ N : n ≤ s } ) for s ∈ N .We will refer to a sequence space L on N as being subsymmetric if foreach increasing function φ : N → N , the operator S φ : L → L definedby ( a n ) ∞ n =1 ( b n ) ∞ n =1 , where b k = ( a n if k = φ ( n ) , S φ is an isometry for every one-to-onemap φ , L will be said to be symmetric .Given a sequence space L on J , and a family ( X j , k · k X j ) j ∈J of(possibly repeated) quasi-Banach spaces with moduli of concavity uni-formly bounded, the space M j ∈J X j ! L = ( f = ( f j ) j ∈J ∈ Y j ∈J X j : (cid:13)(cid:13) ( k f j k X j ) n ∈J (cid:13)(cid:13) L < ∞ ) is a quasi-Banach space with the quasi-norm k f k = (cid:13)(cid:13) ( k f j k X j ) n ∈J (cid:13)(cid:13) . Let ( Y j ) j ∈J be another collection of (possibly repeated) quasi-Banachspaces. If for each j ∈ J , the map T j : X j → Y j is a bounded linearoperator and M := sup j ∈J k T j k < ∞ , then the linear operator T : M j ∈J X j ! L → M j ∈J Y j ! L , ( f j ) j ∈J ( T j ( f j )) j ∈J is bounded with k T k ≤ M .The dual space L ∗ of a minimal sequence space on J can be isomet-rically identified with a sequence space on J . Thus, the dual space of (cid:16)L j ∈J X j (cid:17) L can be isometrically identified with (cid:16)L j ∈J X ∗ j (cid:17) L ∗ .For each k ∈ J let L k : X k → ( L j ∈J X j ) L be the canonical embed-ding. If there is a constant K such that, for each j ∈ J , X j = ( x j,n ) n ∈N j is a K -unconditional basic sequence, then the sequence M j ∈J X j ! L = ( L j ( x j,n )) n ∈N j , j ∈J F. ALBIAC AND J. L. ANSORENA is a K -unconditional basic sequence of ( L j ∈J X j ) L . If X j is normalizedfor all j ∈ J , so is ( L j ∈J X j ) L . If X j is a basis of X j for all j ∈ J and L is minimal, then (cid:16)L j ∈J X j (cid:17) L is a basis of X = ( L j ∈J X j ) L whose dual basis is (cid:16)L j ∈J X ∗ j (cid:17) L ∗ via the aforementioned identificationbetween X ∗ and (cid:16)L j ∈J X ∗ j (cid:17) L ∗ .If J is finite and L = ℓ ∞ ( J ) we set L j ∈J X j = ( L j ∈J X j ) L and L j ∈J X = ( L j ∈J X j ) L . If X j = X for all j ∈ J , we set L ( X ) =( L j ∈J X j ) L . Similarly, if X j = X for all j ∈ J , we set L ( X ) =( L j ∈J X j ) L . Finally, given s ∈ N , we put X s = ℓ s ∞ ( X ) and X s = ℓ s ∞ ( X ).Suppose that X = ( x n ) n ∈N and Y = ( y n ) n ∈N are families of vectorsin quasi-Banach spaces X and Y , respectively. Let C ∈ (0 , ∞ ). Wesay that X C - dominates Y if there is a linear map T from [ X ] into Y with T ( x n ) = y n for all n ∈ N and k T k ≤ C . If T is an isomorphicembedding with max {k T k , k T − k} ≤ C ∈ [1 , ∞ ), X and Y are saidto be C - equivalent . We say that X is permutatively C -equivalent to afamily Y = ( y m ) m ∈M in Y , and we write X ∼ C Y , if there is a bijection π : N → M such that X and ( y π ( n ) ) n ∈N are C -equivalent. A subbasis of an unconditional basis ( x n ) n ∈N is a family ( x n ) n ∈M for some subset M of N .The symbol Y ⊂ ∼ C X will mean that the unconditional basic sequence Y is C -equivalent to a permutation of a subbasis of the unconditionalbasis X . In all cases, if the precise constants are irrelevant, we simplydrop them from the notation.A sequence ( x n ) n ∈N in a quasi-Banach space X said to be semi-normalized if 0 < a := inf n ∈N k x n k ≤ b := sup n ∈N k x n k < ∞ . If a = b = 1 we say that ( x n ) n ∈N is normalized .Given an unconditional basic sequence X = ( x n ) n ∈N and non-zeroscalars ( a n ) n ∈N , the rescaled basic sequence ( a n x n ) n ∈N is equivalentto X if and only if ( a n ) n ∈N is semi-normalized. Thus, the proper-ties related to the uniqueness of unconditional bases in quasi-Banachspaces must be stated in terms of normalized (or, equivalently, semi-normalized) basic sequences. We say that a quasi-Banach space X hasa unique unconditional basis up to equivalence and permutation (UTAPunconditional basis for short) if it has a normalized basis X and anyother normalized basis is permutatively equivalent to X . Other more specific terminology will be introduced in context whenneeded. 3.
Preliminary results
Our approach to the uniqueness of unconditional basis problem in infi-nite direct sums of quasi-Banach spaces will rely on an amalgamationof a set of techniques, most of which are specific to the non-locally con-vex case. In this preparatory section we present the properties and thedifferent methods that will be used in the proofs of our main results inSections 4 and 5.The earliest applications of combinatorial methods to the uniquenessof basis problem can be found in the work of Mitjagin in the early 1970’s[31, 32], but it was W´ojtowicz who gave in 1988 a precise formulation ofthe so-called Schr¨oder-Bernstein principle for unconditional bases (see[37, Corollary 1]).
Theorem 3.1 (Schr¨oder-Bernstein principle for unconditional bases) . Let X and Y be unconditional bases of quasi-Banach spaces X and Y ,respectively. Suppose that X ⊂ ∼ Y and Y ⊂ ∼ X . Then X ∼ Y . Wojtaszczyk rediscovered independently ten years later, in 1997, theidea of using a combinatorial argument in his study of the uniqueness ofunconditional basis of H p ( T ) for 0 < p < Theorem 3.2 ( Hall’s Marriage Lemma) . Let N be a set and ( N i ) i ∈ I be a family of finite subsets of N . Suppose that | F | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)[ i ∈ F N i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for all F ⊆ I finite. Then there is a one-to-one map φ : I → N with φ ( i ) ∈ N i for every i ∈ I . We next enunciate a simple lemma, whose straightforward proof weomit.
Lemma 3.3.
Let L be a sequence space on a countable set J , andfor j ∈ J let X j and Y j be quasi-Banach spaces with moduli of con-cavity uniformly bounded by κ . Suppose that for each j ∈ J , X j is anormalized K -unconditional basic sequence of X j and that Y j is an un-conditional basic sequence of Y j which is C -equivalent to X j , where K F. ALBIAC AND J. L. ANSORENA and C are constants independent of j .Then the semi-normalized uncon-ditional basic sequence ( ⊕ j ∈J Y j ) L of ( ⊕ j ∈J Y j ) L is C -equivalent to thenormalized unconditional basic sequence ( L j ∈J X j ) L of ( L j ∈J X j ) L . Our first result provides sufficient conditions for an infinite directsum of unconditional bases to be equivalent to its square.
Lemma 3.4.
Let L be a sequence space on a countable set J . For each j ∈ J let X j be a normalized K -unconditional basis of a quasi-Banachspace X j with modulus of concavity bounded above by κ , where κ and K are constants independent of j . Suppose that one the the followingconditions holds:(a) There is a constant C such that X j ∼ C X j for all j ∈ J .(b) L is lattice isomorphic to L , and X j = Y for all j ∈ J and someunconditional basis Y .(c) L is subsymmetric, and there is constant C such that, for each j ∈ J , X j ⊂ ∼ C X k for infinitely many values of k ∈ J = N .Then the basis X = ( L j ∈J X j ) L is equivalent to a permutation of itssquare.Proof. The unconditional basis X is equivalent to a permutation of( L j ∈J X j ) L . Thus, if (a) holds, applying Lemma 3.3 yields X ∼ X .The basis X is also equivalent to a permutation of ( L j ∈J X j ) L .Therefore, in the cases (b) and (c), since L is lattice isomorphic to L , X is equivalent to a permutation of X ′ := ( L j ∈N X φ ( j ) ) L for somemap φ : N → N . If (b) holds, X φ ( j ) = X j for all j ∈ N so that X ′ = X . Finally, assume that (c) holds. Then, we recursively constructan increasing map ψ : N → N such that X φ ( j ) ⊂ ∼ C X ψ ( j ) . By Lemma 3.3, X ′ ⊂ ∼ X ′′ := (cid:0)L ∞ n =1 X ψ ( n ) (cid:1) L . By subsymmetry, X ′′ is isometricallyequivalent to a subbasis of X . Hence, by Theorem 3.1, X ∼ X . (cid:3) The Cassaza-Kalton Paradigm extended.
In a couple of pa-pers of classical elegance (see [15, 16]), Casazza and Kalton cruciallyused the lattice structure induced by an unconditional basis on a Ba-nach space to provide a much shorter proof than the original one ofthe uniqueness of unconditional basis UTAP of c ( ℓ ). Of course, thesetechniques were not yet available when Bourgain et al. wrote theirAMS Memoir [13], otherwise the proofs of their aforementioned resultswould have been considerably simpler.Cassaza and Kalton’s methods were transferred to the setting ofquasi-Banach lattices and put into practice in [7] to obtain the unique-ness of unconditional basis UTAP in the spaces ℓ ( ℓ p ) and ℓ p ( ℓ ) for0 < p <
1, and in [10] to give a much shorter proof than the original one of the uniqueness of unconditional basis of ℓ p ( c ) for 0 < p < L -convexity and anti-Euclidean spaces, which we recallnext for the convenience of the reader.A quasi-Banach lattice X is said to be L -convex if there is 0 < ε < ε k f k ≤ max ≤ i ≤ k k f i k whenever f and ( f i ) ki =1 in X satisfy (1 − ε ) kf ≥ P ki =1 f i and 0 ≤ f i ≤ f for every i = 1, . . . , k . We say that a family ( X j ) j ∈J of quasi-Banachlattices is L -convex if there is ε > X j is L -convexwith constant ε for every j ∈ J . Kalton [22] showed that a quasi-Banach lattice X is L -convex if and only if it is p -convex for some p >
0, that is, for some constant C and all f , . . . , f k in X we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X i =1 | f i | p ! /p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C k X i =1 k f i k p ! /p . (3.1)The element ( P ki =1 | f i | p ) /p of X is defined via the procedure outlinedin [30, pp. 40-41]. The optimal constant in (3.1) will be denoted by M p ( X ).Quantitatively, if X is L -convex with constant ε , there exists r > C p )
C ≥ M p ( L ) ≤ C and M p ( X j ) ≤ C for all j ∈ J , then M p ( X ) ≤ C .A quasi-Banach space X is then called natural if it is isomorphic toa subspace of an L -convex quasi-Banach lattice. Most quasi-Banachspaces arising in analysis are natural. However, it should be pointedout that there are non-natural spaces with an unconditional basis [23].It is known [22] that any lattice structure on a natural quasi-Banachspace is L -convex. Thus, once we make sure that a quasi-Banach space X has a lattice structure, the notions of L -convexity and naturalitybecome equivalent. Our results will apply to those natural spaces where the lattice struc-ture is induced by an unconditional basis. In such spaces any uncon-ditional basis induces an L -convex lattice structure; then many of thestandard techniques of Banach lattice theory can be employed in thissetting. For most applications it is easy to verify that the spaces of in-terest are natural either by by showing that some given unconditionalbasis is already p -convex for some p > L -convex lattices. Definition 3.5.
A family ( X j ) j ∈J of unconditional bases of quasi-Banach spaces ( X j ) j ∈J is said to be L -convex if there are constants K ≥ < ε < X j is K -unconditional and it inducesan L -convex lattice structure on X j with constant ε for all j ∈ J .Notice that if ( X j ) j ∈J is an L -convex family of unconditional basesof quasi-Banach spaces ( X j ) j ∈J , then the modulus of concavity of thespace X j is uniformly bounded. Moreover, if L is an L -convex se-quence space over J , then ( L j ∈J X j ) L is an unconditional basis of thequasi-Banach space ( L j ∈J X j ) L which induces a structure of L -convexlattice.A Banach space X is said to be anti-Euclidean if it does not containuniformly complemented copies of finite-dimensional Hilbert spaces.As for L -convexity, to deal with families of quasi-Banach spaces weneed a more quantitative definition. Definition 3.6.
A family ( X j ) j ∈J of Banach spaces is said to be anti-Euclidean if for every R ∈ (0 , ∞ ) there is k ∈ N such that k S k k T k ≥ R whenever j ∈ J and S : ℓ k → X j , T : X j → ℓ k are linear operators with T ◦ S = Id ℓ k .By the principle of local reflexivity, a family ( X j ) j ∈J of Banachspaces is anti-Euclidean if and only if ( X ∗ j ) j ∈J is. The most natu-ral and important examples of anti-Euclidean spaces are c and ℓ . Letus bring up a result by Casazza and Kalton. Theorem 3.7 ([16, Proposition 2.4]) . Suppose that the countable fam-ily ( X j ) j ∈J of Banach spaces is anti-Euclidean. Then the Banach space ( L j ∈J X j ) ℓ is anti-Euclidean. Note that, although Definition 3.6 makes sense for quasi-Banachspaces, as a matter of fact we only state it (and will use it) for the“closest” Banach spaces to the quasi-Banach spaces we study, i.e., theirBanach envelopes. Formally speaking, the
Banach envelope of a quasi-Banach space X consists of a Banach space b X together with a linearcontraction E X : X → b X , called the envelope map of X , satisfying the following universal property: for every Banach space Y and every linearcontraction T : X → Y there is a unique linear contraction b T : b X → Y such that b T ◦ E X = T . The Banach envelope of a quasi-Banach spacecan be effectively constructed from the Minkowski functional of co( B X ).This construction shows that E X (co( B X )) is a dense subset of B b X . Wesay that a Banach space Y is the Banach envelope of X via the map J : X → Y if the associated map b J : b X → Y is an isomorphism.The Banach envelope of a minimal sequence space is isometricallyisomorphic to a minimal sequence space via the inclusion map (see[3, Proposition 10.9]). We will need the following result. Proposition 3.8.
Let L be a minimal sequence space on J . Supposethat X j is a quasi-Banach space with modulus of concavity bounded bya uniform constant κ for all j ∈ J . Then the Banach envelope of X = ( L j ∈J X j ) L is isometrically isomorphic to Y = ( L j ∈J c X j ) b L viathe map f = ( f j ) j ∈J J ( f ) = ( E X j ( f j )) j ∈J . Proof.
Since J defines a linear contraction from X into Y , it suffices toprove that J (co( B X )) is a dense subset of B Y . Let f = ( f j ) j ∈J ∈ B Y and ε >
0. For each j ∈ J set g j = f j / k f j k if f j = 0 and g j = 0otherwise. Since g j ∈ B c X j , for each j ∈ J there is h j ∈ co( B X j ) suchthat k g j − E X j ( h j ) k ≤ ε/
2. Put Γ = ( γ j ) j ∈J , where γ j = k f j k for j ∈ J . Since Γ ∈ B b L , there is Λ ∈ co( B L ) such that k Γ − Λ k b L ≤ ε/ λ j ) j ∈J , we have that h := ( λ j h j ) j ∈J ∈ co( B X ). Therefore, if g = ( λ j g j ) j ∈J , k f − E ( h ) k ≤ k f − g k + k g − E ( h ) k = k Γ − Λ k b L + k ( λ j k g j − E j ( h j ) k ) j ∈J k b L ≤ ε ε k Λ k b L ≤ ε. (cid:3) In most cases, the proof of the uniqueness of unconditional basis in agiven Banach (or quasi-Banach) space also sheds light onto the uncon-ditional structure of its complemented subspaces with an unconditionalbasis. A sequence Y = ( y m ) m ∈M in a quasi-Banach space X is said tobe complemented if its closed linear span Y = [ Y ] is a complementedsubspace of X , i.e., there is a bounded linear map P : X → Y with P | Y = Id Y . An unconditional basic sequence Y = ( y m ) m ∈M is com-plemented in X if and only if there exists a sequence Y ∗ = ( y ∗ m ) m ∈M in X ∗ such that y ∗ m ( u n ) = δ m,n for all ( m, n ) ∈ M and there is a bounded linear map P : X → X given by P ( f ) = P [ Y , Y ∗ ]( f ) = X m ∈M y ∗ m ( f ) y m , f ∈ X, (3.2)in which caseΓ[ Y , Y ∗ ] := sup ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈ M y ∗ m ( f ) y m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) : M ⊆ M , f ∈ B X ) < ∞ . We will refer to Y ∗ as a sequence of projecting functionals for Y .To understand the simplifications derived from taking into accountthe lattice structure induced by an unconditional basis X on the entirespace X , we must look at the supports of Y and Y ∗ with respect to X . Definition 3.9.
Let X be a quasi-Banach space with an unconditionalbasis X . We say that an unconditional basic sequence Y = ( y m ) m ∈M is well complemented in X if it is complemented in X and there is asequence Y ∗ = ( y ∗ m ) m ∈M of projecting functionals for Y such that:(i) supp( y ∗ m ) ⊆ supp( y m ) for all m ∈ M , and(ii) (supp( y m )) m ∈M is a pairwise disjoint family consisting of finitesets.In this case, we say that Y ∗ is a sequence of good projecting functionals for Y . If Γ[ Y , Y ∗ ] ≤ C we will say that Y is well C -complemented andthat Y ∗ are good C -projecting functionals. Remark . Note that a subbasis of a well C -complemented basicsequence ( y m ) m ∈M is a a well C -complemented basic sequence. In par-ticular, if ( y ∗ m ) m ∈M are good C -projecting functionals, k y m k k y ∗ m k ≤ C for all m ∈ M .The following definition identifies and gives relief to an unstatedfeature shared by some unconditional bases. Examples of such basescan be found, e.g., in [9, 15, 21], where the property naturally arises inconnection with the problem of uniqueness of unconditional basis. Definition 3.11.
A normalized unconditional basis X = ( x n ) n ∈N of aquasi-Banach space will be said to be universal for well complementedblock basic sequences if for every normalized well complemented basicsequence Y = ( y m ) m ∈M of X there is a map π : M → N such that π ( m ) ∈ supp( y n ) for every m ∈ M , and Y is equivalent to the rear-ranged subbasis ( x π ( m ) ) m ∈M of X . In the case when there is a function η : [1 , ∞ ) → [1 , ∞ ) such that Y is η ( C )-equivalent to ( x π ( m ) ) m ∈M of X whenever Y is well C -complemented, we say that X is uniformlyuniversal for well complemented block basic sequences (with function η ). Thus, the following theorem summarizes what can be rightfully calledthe “Casazza-Kalton paradigm” to tackle the uniqueness of uncondi-tional basis problem extended to quasi-Banach lattices. To be able toprove it in this optimal form (even for locally convex spaces) requiredthe very recent solution in the positive of the “canceling squares” prob-lem (see [2]).
Theorem 3.12 (see [2, Theorem 3.9]) . Let X be a quasi-Banach spacewith a normalized unconditional basis X . Suppose that:(i) The lattice structure induced by X in X is L -convex;(ii) The Banach envelope of X is anti-Euclidean;(iii) X is universal for well complemented block basic sequences; and(iv) X ∼ X .Then X has a UTAP unconditional basis. The peaking property.
Another technique that has become cru-cial to determine the uniqueness of unconditional basis in quasi-Banachspaces is the “large coefficient technique.” It was introduced by Kaltonin [21] to prove the uniqueness of unconditional basis in nonlocallyconvex Orlicz sequence spaces ℓ F . Kalton called a complemented basicsequence ( y n ) in ℓ F inessential ifinf n sup k | y ∗ n ( x k ) | | x ∗ k ( y n ) | > , and proved that if ( y n ) is inessential then it is equivalent to the canon-ical basis ( x k ) of ℓ F .Kalton’s ideas were extended to the general framework of quasi-Banach lattices in [24]. Here we reformulate this property and regardit as a feature of the unconditional basis ( x n ) of the space instead ofthe complemented basic sequence ( y n ). Definition 3.13.
An unconditional basis X = ( x n ) n ∈N of a quasi-Banach space X will be said to have the peaking property if for everywell complemented basic sequence Y = ( y m ) m ∈M with respect to X there is a sequence ( y ∗ m ) m ∈M of good projecting functionals such that c := inf m ∈M sup n ∈N | y ∗ m ( x n ) | | x ∗ n ( y m ) | > . (3.3)In the case when there is a function γ : [1 , ∞ ) → [1 , ∞ ) such that γ ( C ) ≥ /c whenever Y is well C -complemented, we say that X hasthe uniform peaking property (with function γ ).The proof of Proposition 3.15 below relies on the following reductionlemma which will be used as well in Section 4. Lemma 3.14 (cf. [1, Lemma 3.1]) . Let Y = ( y m ) m ∈M be a wellcomplemented basic sequence with respect to an unconditional basis X = ( x n ) n ∈N of a quasi-Banach space X , and let ( y ∗ m ) m ∈M be a se-quence of good projecting functionals for Y . Suppose U = ( u m ) m ∈M and ( u ∗ m ) m ∈M are sequences in X and X ∗ respectively such that:(i) | x ∗ n ( u m ) | ≤ D | x ∗ n ( y m ) | for all ( n, m ) ∈ N × M ,(ii) | u ∗ m ( x n ) | ≤ D | y ∗ m ( x n ) | for all ( n, m ) ∈ N × M , and(iii) | u ∗ m ( u m ) | ≥ /D for all m ∈ M ,for some positive constants D , D and D . Then U is a well com-plemented basic sequence equivalent to Y . Quantitatively, if Y is well C -complemented, and X is K -unconditional, then:(a) The sequence U is well B -complemented with good B -projectingfunctionals ( λ m u ∗ m ) m ∈M , where B = CD D D K and λ m =1 / u ∗ m ( u m ) ; and(b) The basic sequence Y CD K -dominates U ; and(c) the basic sequence U CD D K -dominates Y .Proof. The proof follows the steps of the proof of [1, Lemma 3.1], keep-ing track of the constants involved. (cid:3)
Proposition 3.15 (cf. [2, Proposition 3.3]) . Let X be a quasi-Banachspace with a normalized unconditional basis X . If X has the peak-ing property, then X is universal for well complemented block basicsequences. Moreover, if X is K -unconditional and has the uniformpeaking property with function γ , then X is uniformly universal forwell-complemented block basic sequences with function C KCγ ( C ) .Proof. Go through the proof of [2, Proposition 3.3] with Lemma 3.14in mind, paying attention to the constants involved. (cid:3)
The following lemma relies on Lemma 3.14. Given a family X =( x n ) n ∈N in a quasi-Banach space X and A ⊆ N finite, we will use thenotation A [ X ] = X n ∈ A x n . Lemma 3.16 (cf. [7, Lemma 4.1]) . Suppose X is a normalized K -unconditional basis of a quasi-Banach space X with dual basis X ∗ .Assume that X D -dominates the unit vector system of ℓ . If B =4 C DK , then for every normalized C -complemented basic sequence U in X there is a well B -complemented basic sequence Y = ( y m ) m ∈M in X such that:(i) supp( y m ) ⊆ supp( u m ) for all m ∈ M ;(ii) Y is E -equivalent to U , where E = 2 CK max { C, D } ; and (iii) ( supp( y m ) [ X ∗ ]) m ∈M is a family of good B -projecting functionalsfor Y .Proof. Just go over the lines of the proof of [2, Lemma 3.6] payingattention to the constants involved. (cid:3)
Strongly absolute bases.
Strong absoluteness was identified byKalton, Ler´anoz, and Wojtaszczyk in [24] as the crucial differentiatingfeature of unconditional bases in quasi-Banach spaces in their investi-gation of the uniqueness of unconditional bases. One could say thatstrongly absolute bases are “purely nonlocally convex” bases, in thesense that if a quasi-Banach space X has a strongly absolute basis,then its unit ball is far from being a convex set and so X is far frombeing a Banach space. Although the term strongly absolute for a basiswas coined in [24], here we work with a slightly different but equivalentdefinition. Definition 3.17.
An unconditional basis X = ( x n ) n ∈N of a quasi-Banach space X is strongly absolute if for every constant R >
C > X n ∈N | x ∗ n ( f ) | k x n k ≤ max (cid:26) C sup n ∈N | x ∗ n ( f ) | k x n k , k f k R (cid:27) , f ∈ X. (3.4)If α : (0 , ∞ ) → (0 , ∞ ) is such that (3.4) holds with C = α ( R ) for every0 < R < ∞ , we say that X is strongly absolute with function α .Note that if we rescale a strongly absolute basis we obtain a stronglyabsolute basis with the same function. Note also that a normalizedunconditional basis X = ( x n ) n ∈N is strongly absolute with function α if and only if k f k < R X n ∈N | x ∗ n ( f ) | = ⇒ X n ∈N | x ∗ n ( f ) | ≤ α ( R ) max n ∈N | x ∗ n ( f ) | . If X = ( x n ) n ∈N is a strongly absolute basis with function α of a quasi-Banach space X , the normalized basis ( x n / k x n k ) n ∈N D -dominates theunit vector basis of ℓ ( N ), where D = D ( α ) = inf R> max (cid:26) α ( R ) , R (cid:27) . Roughly speaking a normalized (or semi-normalized) unconditional ba-sis is strongly absolute if and only if it dominates the unit vector basisof ℓ , and whenever the ℓ -norm and the quasi-norm of a vector arecomparable then so are the ℓ ∞ -norm and the ℓ -norm of its coordi-nates. Adding combinatorial arguments to the methods from [24] enabledWojtaszczyk to prove the following criterion for spaces with a stronglyabsolute basis. Needless to say, he could not count on the Casazza-Kalton paradigm since it had not been discovered yet.
Theorem 3.18 (See [36, Theorem 2.12]) . Let X be a natural quasi-Banach space with a strongly absolute unconditional basis ( x n ) n ∈N . As-sume also that X is isomorphic to some of its cartesian powers X s , s ≥ . Then all normalized unconditional bases of X are permutativelyequivalent. For further reference, we record an elementary lemma.
Lemma 3.19.
Let X and Y be normalized unconditional bases of quasi-Banach spaces X and Y respectively. Suppose that X is strongly abso-lute with function α and that Y D -dominates X . Then Y is stronglyabsolute with function Dα . The following proposition guarantees the strongly absoluteness ofinfinite direct sums of strongly absolute bases. Some applications inSection 6 will relay on it as we shall see.
Proposition 3.20.
Let L be a sequence space on a set J with stronglyabsolute basis. For each j ∈ J let X j be a K -unconditional basis ofa quasi-Banach space with modulus of concavity at most κ , where κ and K are constants independent of j . Suppose that there is α suchthat X j is strongly absolute with function α for all j ∈ J . Then X := ( L j ∈N X j ) L is a strongly absolute unconditional basis of X :=( L j ∈N X j ) L . Moreover, if β is a strongly absolute function for the unitvector system of L , then the map R γ ( R ) := β ( RD ( α )) α ( Rβ ( RD ( α ))) , < R < ∞ , is a strongly absolute function for X .Proof. Without lost of generality we assume that X j is normalized forall j ∈ J so that X is normalized too. For each j ∈ J let F j be thecoefficient transform with respect to X j . Let β be a strongly absolutefunction for L . Pick f = ( f j ) j ∈J ∈ X and R ∈ (0 , ∞ ) such that k f k = k ( k f j k ) j ∈J k L ≤ R k ( kF j ( f j ) k ) j ∈J k . Since, by unconditionality, k ( kF j ( f j ) k ) j ∈J k L ≤ D ( α ) k ( k f j k ) j ∈J k L , we obtain kF ( f ) k = k ( kF ( f j ) k ) j ∈J k ≤ β ( RD ( α )) sup j ∈J kF j ( f j ) k . Let k ∈ J be such that kF k ( f k ) k = sup j ∈J kF j ( f j ) k . By uncondi-tionality, k f k k ≤ k f k L ≤ Rβ ( RD ( α )) sup j ∈J kF j ( f j ) k = Rβ ( RD ( α )) kF k ( f k ) k , so that,sup j ∈J kF j ( f j ) k = kF k ( f k ) k ≤ α ( Rβ ( RD ( α ))) kF k ( f k ) k ∞ . Since kF ( f ) k ∞ = sup j kF j ( f j ) k ∞ , we obtain kF ( f ) k ≤ γ ( R ) kF ( f ) k ∞ . (cid:3) Infinite L -sums of quasi-Banach spaces, where L is asequence space with a strongly absolute basis Our first theorem in this section uses the previous ingredients and thelanguage introduced in Section 3 to provide, in particular, an extensionof [7, Theorem 4.5], which established the uniqueness of unconditionalbasis UTAP in the spaces ℓ p ( ℓ ) for 0 < p < Theorem 4.1.
Let L be a sequence space on a set J with a stronglyabsolute basis. For each j ∈ J , let X j be a normalized K -unconditionalbasis of quasi-Banach space X j with modulus of concavity at most κ ,where K and κ are independent of j . Suppose that there is a func-tion η : [1 , ∞ ) → [1 , ∞ ) such that X j is uniformly universal for wellcomplemented block basic sequences with function η for all j ∈ J .Then the unconditional basis ( L j ∈J X j ) L of the infinite direct sum X := ( L j ∈J X j ) L is uniformly universal for well complemented blockbasic sequences.Proof. We isometrically identify X ∗ with V := ( ⊕ j ∈J X ∗ j ) L ∗ via thenatural dual pairing h· , ·i : V × X → F . For each j ∈ J , let L j : X j → X and L ′ j : X ∗ j → V be the natural ‘inclusion’ maps, and let T j : X → X j be the natural projection. Set X j = ( x j,n ) n ∈N j . Let C ∈ [1 , ∞ ) and let Y = ( y m ) m ∈M be a normalized well C -complemented basis sequencewith good C -projecting functionals Y ∗ = ( y ∗ m ) m ∈M . Let ( v m ) m ∈M thecorresponding sequence in V via the above described dual mapping.Set y m = ( y j,m ) j ∈J and v m = ( y ∗ j,m ) j ∈J for each m ∈ M . Set also f m = ( y ∗ j,m ( y j,m )) ∈J , m ∈ M . For m ∈ M , we have1 = y ∗ m ( y m ) = | y ∗ m ( y m ) | = |h v m , y m i| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ∈J y ∗ j,m ( y j,m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k f m k and k f m k L ≤ (cid:13)(cid:13) ( k y ∗ j,m k k y j,m k ) j ∈J (cid:13)(cid:13) L ≤ sup j ∈J k y ∗ j,m k k ( k y j,m k ) j ∈J k L = k v m k k y m k≤ k y ∗ m k k y m k≤ C. Hence, if the unit vector system of L is strongly absolute with function α , we have k f m k ∞ ≥ α ( C ) , m ∈ M . Therefore, there is a map φ : M → J such that (cid:12)(cid:12)(cid:10) L ′ φ ( m ) ( y ∗ φ ( m ) ,m ) , L φ ( m ) ( y φ ( m ) ,m ) (cid:11)(cid:12)(cid:12) = | y ∗ φ ( m ) ,m ( y φ ( m ) ,m ) | ≥ α ( C )for all m ∈ M . By Lemma 3.14, the sequence ( L φ ( m ) ( y φ ( m ) ,m )) m ∈M is E -equivalent to Y , where E = α ( C ) CK . We have, in particular,1 E k y φ ( m ) ,m k ≤ , m ∈ M . Set B = E CK and E ′ = ECK . Applying again Lemma 3.14 givesthat U = ( u m ) m ∈M := ( L φ ( m ) ( y φ ( m ) ,m ) / k y φ ( m ) ,m k ) m ∈M is a normalized well B -complemented basic sequence E ′ -equivalent to Y . For each j ∈ J put M j = { m ∈ M : φ ( m ) = j } . Composing the projections from X onto X associated to the well com-plemented basic sequence ( u m ) m ∈M j with the maps L j and T j we obtainthat Y j = ( y j,m / k y j,m k ) m ∈M j is well B -complemented in X j . By assumption, for each j ∈ J thereis a map ν j : M j → N j such that ν j ( m ) ∈ supp( y j,m ) for all m ∈ M j and Y j is η ( B )-equivalent to ( x j,ν ( m ) ) m ∈M j . By Lemma 3.3, U is η ( B )-equivalent to ( L φ ( m ) ( x φ ( m ) ,ν ( m ) )) m ∈M . (cid:3) We are now ready to obtain the main theoretical result of the section.
Theorem 4.2.
Let L be an L -convex sequence space on a countableset J . Let ( X j ) j ∈J be an L -convex family of normalized unconditionalbases of quasi-Banach spaces ( X j ) j ∈J . Suppose that: (i) X j is uniformly universal for well-complemented block basic se-quences with function η for all j ∈ J ;(ii) the family of Banach envelopes (cid:16) c X j (cid:17) j ∈ J is anti-Euclidean;(iii) The unit vector system of L is strongly absolute; and(iv) one the the following conditions holds:(a) there is constant C such that X j ∼ C X j for all j ∈ J .(b) L is lattice isomorphic to L , and X j = Y for all j ∈ J andsome unconditional basis Y .(c) L is subsymmetric, and there is a constant C such that, foreach j ∈ J , X j ⊂ ∼ C X k for infinitely many values of k ∈ J .Then X = ( L j ∈J X j ) L has a UTAP unconditional basis.Proof. Since the unit vector system of L is strongly absolute, its Banachenvelope is lattice isomorphic to ℓ . By Proposition 3.8, the Banachenvelope of X is isomorphic to (cid:16)L j ∈J c X j (cid:17) ℓ , which is anti-Euclideanby Theorem 3.7. By Theorem 4.1, X = ( L j ∈J X j ) L is (uniformly)universal for well complemented block basic sequences. By Lemma 3.4, X ∼ X . Applying Theorem 3.12 puts an end to the proof. (cid:3) Remark . A variation of the argument used to prove Theorem 4.1gives that, if we replace the hypothesis “ X j is uniformly universal forwell-complemented block basic sequences with function η for all j ∈ J ”with “there is a function γ : [1 , ∞ ) → [1 , ∞ ) such that X j has theuniform peaking property with function γ ”, we obtain that the un-conditional basis ( L j ∈J X j ) L of ( L j ∈J X j ) L has the uniform peak-ing property. In particular, any strongly absolute unconditional basis X = ( x n ) n ∈N of any quasi-Banach space X has the uniform peakingproperty. Let us see a more direct proof of this result. For any f ∈ X and f ∗ ∈ X ∗ we have | f ∗ ( f ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ∈N f ∗ ( x n ) x ∗ n ( f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X n ∈N | f ∗ ( x n ) | | x ∗ n ( f ) | , and, if X is K -unconditional and normalized, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X n ∈N | f ∗ ( x n ) x ∗ n ( f ) x n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ K sup n ∈N | f ∗ ( x n ) | k f k ≤ CK k f ∗ kk f k . Hence, if k f ∗ k k f ∗ k ≤ C | f ∗ ( f ) | , | f ∗ ( f ) | ≤ α ( CK ) sup n ∈N | f ∗ ( x n ) | | x ∗ n ( f ) | , where α is the strongly absolute function of X . We infer that X hasthe uniform peaking property with function C α ( CK ). Infinite ℓ -sums of spaces with strongly absolute bases In this section, we generalize the main result from [11] and solve anexplicit problem raised ten years ago in [11, Remark 3.6]. With hind-sight, and in light of Theorem 3.18, it also sets right [11, Corollary 3.4],whose validity seemed to rely on a wrong set of hypotheses.
Theorem 5.1.
For each j ∈ J , let X j be a normalized K -unconditionalbasis of a quasi-Banach space X j with modulus of concavity at most κ .Suppose that there is α such that X j is strongly absolute with function α for all j ∈ J . Then the unconditional basis X = ( L j ∈J X j ) ℓ of ( L j ∈J X j ) ℓ is uniformly universal for well-complemented block basicsequences. The proof of Theorem 5.1 will be shortened considerably after takingcare of the following lemma.
Lemma 5.2.
For each j ∈ J , let X j = ( x j,n ) n ∈N j be a normalized K -unconditional basis of a quasi-Banach space X j with modulus of con-cavity at most κ and let L j : X j → X := ( ⊕ j ∈J X j ) ℓ be the canonicalembedding. For ( j, n ) ∈ N := ∪ j ∈J { j } × N j denote x j,n = L j ( x j,n ) ,so that X := ( ⊕ j ∈J X j ) ℓ = ( x j,n ) ( j,n ) ∈N . For each j ∈ J , let X ∗ j =( x ∗ j,n ) n ∈N j denote the dual basis of X j , and let X ∗ = ( x ∗ j,n ) ( j,n ) ∈N bethe dual basis of X . Suppose Y = ( y m ) m ∈M is a normalized well C -complemented normalized basic sequence with respect to the normalizedunconditional basis X of X for which ( y ∗ m ) m ∈M = ( supp( y m ) [ X ∗ ]) m ∈M is a family of good C -projecting functionals. Put y m = ( y j,m ) j ∈J andset J m = { j ∈ J : y j,m = 0 } . Then:(a) If X j D -dominates the unit vector system of ℓ for all j ∈ J , ( y m ) m ∈M D -dominates the unit vector system of ℓ ( M ) .(b) If | M | ≤ | ∪ m ∈ M J m | for every M ⊆ M finite, there is a one-to-onemap π : M → N such that the rearranged subbasis ( x π ( m ) ) m ∈M of X is isometrically equivalent to the unit vector system of ℓ and C -dominates Y .(c) If(i) X j is strongly absolute with function α for every j ∈ J , and(ii) there is M ⊆ M finite and nonempty such that | ∪ m ∈ M J m | < | M | ,for every R ∈ (0 , ∞ ) we have ∆ := sup (cid:8) | x ∗ j,n ( y j,m ) | : m ∈ M , j ∈ J , n ∈ N j (cid:9) ≥ R − CR α ( R ) . Proof. (a) The basis X D -dominates the unit vector system of ℓ ( N ).That is, X ( j,n ) ∈N | x ∗ j,n ( f ) | ≤ D k f k , f ∈ X. For ( a m ) m ∈M ∈ c ( M ), write f = P m ∈M a m y m . Then, X m ∈M | a m | = X m ∈M | y ∗ m ( f ) | = X m ∈M X ( j,n ) ∈ supp( y m ) (cid:12)(cid:12) x ∗ j,n ( f ) (cid:12)(cid:12) = X ( j,n ) ∈N (cid:12)(cid:12) x ∗ j,n ( f ) (cid:12)(cid:12) ≤ D k f k . (cid:3) (b) By Theorem 3.2, there is a one-to-one map φ : M → J such that y φ ( m ) ,m = 0 for all m ∈ M . Thus, there is ν : M → ∪ j ∈J N j such that ν ( m ) ∈ N φ ( m ) for all m ∈ M and such that x ∗ φ ( m ) ,ν ( m ) ( y m ) = 0. Define π : M → N by π ( m ) = ( φ ( m ) , ν ( m )) for all m ∈ M . Let T j be thecanonical projection of X onto X j . Since, given m ∈ M , T j ( x π ( m ) ) = 0for at most one j ∈ J , for every ( a m ) m ∈M ∈ c ( M ) we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈M a m x π ( m ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = X j ∈J (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈M a m T j ( x π ( m ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = X j ∈J X m ∈M | a m | k T j ( x π ( m ) ) k = X m ∈M | a m | X j ∈J k T j ( x π ( m ) ) k = X m ∈M | a m | k x π ( m ) ) k = X m ∈M | a m | . Let P = P [ Y , Y ∗ ] be the projection defined in (3.2). If ( j, n ) ∈ sup( y m ) for some m ∈ N , P ( x j,n ) = X m ′ ∈M y ∗ m ′ ( x j,n ) y m ′ = X m ′ ∈M δ m,m ′ y m ′ = y m . Hence, if f = P m ∈M a m y m , k f k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P X m ∈M a m x π ( m ) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈M a m x π ( m ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . (c) Note that X j D ( α )-dominates the unit vector system of ℓ ( N j )for all j ∈ J . Pick M minimal with | ∪ m ∈M J m | < |M | < ∞ . Since J m = ∅ for all m ∈ M we have |M | ≥
2. Pick m ∈ M arbitraryand set M = M \ { m } . By Lemma 5.2 (b), the unit vector system of ℓ ( M ) C -dominates the finite basis ( y m ) m ∈ M . If we set J = ∪ m ∈ M J m , | M | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m ∈ M X ( j,n ) ∈N x ∗ j,n ( y m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ∈ J X n ∈N j x ∗ j,n X m ∈ M y j,m !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X j ∈ J X n ∈N j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ∗ j,n X m ∈ M y j,m !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X j ∈ J max ( α ( R ) sup n ∈N j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ∗ j,n X m ∈ M y j,m !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , R (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈ M y j,m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)) ≤ X j ∈ J max ( α ( R )∆ , R (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈ M y j,m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)) ≤ | J | α ( R )∆ + 1 R X j ∈ J (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈ M y j,m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = | J | α ( R )∆ + 1 R (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈ M y m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ | J | α ( R )∆ + CR | M | Since | J | ≤ | ∪ m ∈M J m | ≤ |M | − | M | , we are done. Proof of Theorem 5.1.
Let C ∈ [1 , ∞ ). Pick R > B := 4 C D ( α ) K , E > Rα ( R ) / ( R − C ) and E ′ = 2 CK max { C, D ( α ) } . Let U be a well C -complemented basic sequence in X . By Lemma 3.16, there is a well B -complemented basic sequence Y = ( y m ) m ∈M in X with good B -projecting functionals ( supp( y m ) [ X ∗ ]) m ∈M which is E ′ -equivalent to U . With the terminology of Lemma 5.2, put M = { m ∈ M : | x ∗ j,n ( y j,m ) | ≤ E for all ( j, n ) ∈ N } and M = M \ M . By Lemma 5.2 there is π : M → N such that π ( m ) ∈ supp( y m ) forall m ∈ M and ( y m ) m ∈M D ( α )-dominates and it is B -dominated by( x π ( m ) ) m ∈M . In turn, there is π : M → N such that | x ∗ π ( m ) ( y m ) | > E , m ∈ M . Hence, by Lemma 3.14, ( y m ) m ∈M BK -dominates and it is BKE -dominated by ( x π ( m ) ) m ∈M . We infer that if D = κC max { BK, D ( α ) } = 4 κC D ( α ) K ,D = κK max { BKE, B } = 4 κC D ( α ) K E, and π : M → N is obtained by glueing the functions π and π , U D -dominates and it is D -dominated by ( x π ( m ) ) m ∈M . (cid:3) Theorem 5.3.
Let ( X j ) j ∈J be an L -convex family of normalized un-conditional bases of quasi-Banach spaces ( X j ) j ∈J . Suppose that:(a) X j is strongly absolute with the same function α for all j ∈ J ; and(b) Either:(i) There is a constant C such that X j ∼ C X j for all j ∈ J , or(ii) There is a constant C such that, for each j ∈ J , X j ⊂ ∼ C X k for infinitely many values of k ∈ N .Then the space X = ( L j ∈J X j ) ℓ has a UTAP unconditional basis.Proof. Since X j D ( α )-dominates the unit vector system of ℓ ( N j ) forall j ∈ J , the normalized basis X = ( L j ∈J X j ) ℓ of X D ( α )-dominatesthe unit vector system of ℓ ( N ). Hence the Banach envelope of X isisomorphic to the anti-Euclidean space ℓ ( N ). By Theorem 5.1 andRemark 4.3, X is (uniformly) universal for well complemented blockbasic sequences. By Lemma 3.4, X ∼ X and so applying Theorem 3.12concludes the proof. (cid:3) Applications and Examples
Theorems 4.2 and 5.3, combined with Proposition 3.20, yield a myriadof new examples of quasi-Banach spaces with a UTAP unconditionalbases. In this section we highlight applications only to a sampler ofinfinite direct sums involving classical spaces, but the reader is encour-aged to create their favourite infinite direct sums and use our previousresults to check that they enjoy the property of uniqueness (UTAP) ofunconditional basis. The possibilities for new examples are endless.6.1.
Lorentz Sequence Spaces.
Let w = ( w n ) ∞ n =1 be a sequence ofnon-negative scalars with w > s n ) ∞ n =1 be the primitive weight of w , defined by s n = n X k =1 w k , n ∈ N . Given 0 < p < ∞ and 0 < q ≤ ∞ , the Lorentz sequence space d p,q ( w )consists of all f ∈ c whose non-increasing rearrangement ( a n ) ∞ n =1 sat-isfies k f k p,q, w := ∞ X n =1 ( a n s /pn ) q w n s n ! /q < ∞ , with the usual modification if q = ∞ . If ( s n ) ∞ n =1 is doubling , i.e.,sup m s m /s ⌈ m/ ⌉ < ∞ , then k · k p,q, w is a quasi-norm. In this case, d p,q ( w ) is a symmetric sequence space. Moreover, if q < ∞ , d p,q ( w ) isminimal.If 0 < p = q <
1, the space d p,p ( w ) coincides with the Lorentz spacedenoted as d ( w , p ) by Altshuler in [12] (see also [2, 9, 14]), while d p, ∞ ( w )coincides with the weak Lorentz space denoted as d ∞ ( w , p ) in [14]. If q = ∞ , we will denote by d p, ∞ ( w ) the separable part of d p, ∞ ( w ), i.e.,the closed linear span of c in d p, ∞ ( w ).If 0 < q < r ≤ ∞ , we have d p,q ( w ) ⊆ d p,r ( w ) , (6.1)and for all A ⊂ N with | A | = m , k A k p,q, w ≈ s /pm . (6.2)Thus it could be said that for fixed p and w , the spaces d p,q ( w ) areclose to each other in the sense that all of them share (essentially) thefundamental function of the canonical basis. This is important to betaken into account when considering embeddings (see below).We point out that if 0 < p, q < ∞ and the primitive weight of w ′ is( s p/qn ) ∞ n =1 , then d p,q ( w ) = d q,q ( w ′ ) , (6.3) up to an equivalent norm. Similarly, if w ′ = ( w ′ n ) ∞ n =1 denotes the weightwhose primitive weight is ( s /pn ) ∞ n =1 , d p, ∞ ( w ) = d , ∞ ( w ′ ) . Thus, every sequence Lorentz space d p,q ( w ) can be identified, up toan equivalent quasi-norm, with a Lorentz sequence space d ,q ( w ′ ) for asuitable weight w ′ . The advantages of establishing results concerningsequence Lorentz spaces in terms of the scale of spaces d ,q ( w ), 0
0. Let C ∈ [1 , ∞ ) be such that s n ≤ Cs n for all n ∈ N . Pick α ∈ (0 ,
1) and k ∈ N such that C α ≤ − /k . Then, if r = 2 k , s αrn ≤ rs αn , n ∈ N , < α < α . That is, ( s αn ) ∞ n =1 has the URP and, then, satisfies inequality (6.4) for all0 < α < α . Let w α be the weight whose primitive weight if ( s αn ) ∞ n =1 .By [14, Theorems 2.5.10 and 2.5.11], d p,q ( w αp ) is locally convex forall α < α and q >
1. Note that local convexity is equivalent tolattice 1-convexity. We infer that d ,q ( w ) is lattice r -convex for all0 < r < min { α , q } . (cid:3) Next we tackle the strong absoluteness or the canonical basis ofLorentz sequence spaces.
Proposition 6.2.
Suppose that the primitive weight ( s n ) ∞ n =1 of w =( w n ) ∞ n =1 is doubling.(a) The following are equivalent:(i) d , ∞ ( w ) is continuously included in ℓ .(ii) P ∞ n =1 /s n < ∞ .(iii) The unit vector system is a strongly absolute basis of d , ∞ ( w ) .(b) Let < q < ∞ and let q ′ be its conjugate exponent. Suppose that P ∞ n =1 w − q ′ +1 n s − n < ∞ . Then the unit vector system is a stronglyabsolute basis of d ,q ( w ) .(c) Let < q ≤ .(i) d ,q ( w ) is continuously included in ℓ if and only if inf n s n /n > . Moreover.(ii) if lim n s n /n = ∞ , then the unit vector system is a stronglyabsolute basis of d ,q ( w ) .(iii) if inf n s n /n > and q < , the unit vector system of d ,q ( w ) is uniformly universal for well complemented block basic se-quences.Proof. The implication (iii) ⇒ (i) in (a) is obvious. If f = (1 /s n ) ∞ n =1 ,we have k f k , ∞ , w = 1. This yields (i) ⇒ (ii). To prove (b) and theimplication (ii) ⇒ (iii) in (a) we pick 1 < q ≤ ∞ and 0 < R < ∞ .Choose m = m ( R ) ∈ N such that ∞ X n = m +1 = 1 w q ′ n w n s n ≤ R ) q ′ . Let f ∈ F N and denote by ( a n ) ∞ n =1 its non-increasing rearrangement.By Holder’s inequality, k f k = m X n =1 a n + ∞ X n = m +1 w n a n s n w n s n ≤ m k f k ∞ + 12 R ∞ X n = m +1 a qn s qn w n s n ! /q ≤ m k f k ∞ + 12 R k f k ,q, w . Thus, if k f k ,q, w ≤ R k f k , we obtain k f k ≤ ma = 2 m k f k ∞ .As far as (c) is concerned, the “only if” part in (i) is clear. By(6.1), to prove the converse it suffices to consider the case q = 1. If w ′ = ( w ′ n ) ∞ n =1 is the weight defined by w ′ n = 1 for all n ∈ N , then d , ( w ) ⊆ d , ( w ′ ). Since d , ( w ′ ) = ℓ we are done.(ii) is essentially known (see [33, Lemma 4] and [24, Theorem 2.6]).However, as an explicit proof is not available in the literature, we nextinclude one for the sake of completeness. Again, by Lemma 3.19, itsuffices to consider the case q = 1. Let R ∈ (0 , ∞ ). Choose m ∈ N such that s n ≥ Rn for all n ≥ m + 1. If ( a n ) ∞ n =1 is the non-increasingrearrangement of f , by Abel’s summation formula, ∞ X n =1 a n = ∞ X n =1 ( a n − a n +1 ) n ≤ m X n =1 ( a n − a n +1 ) n + 12 R ∞ X n = m +1 ( a n − a n +1 ) s n ≤ − ma m +1 + m X n =1 a n + 12 R ∞ X n =1 ( a n − a n +1 ) s n ≤ m k f k ∞ + 12 R k f k , , w . Therefore, k f k ≤ m whenever k f k , , w ≤ R k f k .In regards to (iii), we point out that it was proved in [2, Proposi-tion4.2] that d ,q ( w ) has the peaking property. A close look at theproof of this result reveals that, in fact, it has the uniform peakingproperty. Essentially, this is due to the validity of a constructive ver-sion of the proof of [34, Lemma 3.1]. Specifically, there is a function ζ : (0 , ∞ ) → (0 , ∞ ) depending on p and w such that every normalizeddisjointly supported sequence ( y m ) ∞ m =1 with respect to ( e n ) ∞ n =1 withlim inf m k y m k ∞ < ζ ( ε ) has a subsequence that (1 + ε )-dominates theunit vector system of ℓ p . By Proposition 3.15, ( e n ) ∞ n =1 is uniformlyuniversal for block basic sequences. (cid:3) To complement the theoretical contents of this section we shall in-troduce lattice concavity and a quantitative tool from approximationtheory that serves in particular to measure how far an unconditionalbasis is from the canonical ℓ -basis. The main idea is to use embeddingsinto Lorentz sequence spaces to deduce that certain bases are stronglyabsolute. Given a (semi-normalized) unconditional basis X of a quasi-Banachspace X we define its lower democracy function as ϕ l [ X ]( m ) = inf | A |≥ m k A [ X ] k , m ∈ N . If L is a sequence space, ϕ l [ L ] will denote the lower democracy functionof its unit vector system. The quasi-Banach lattice L is said to be q -concave , 0 < q ≤ ∞ , if there is a nonnegative constant C such that k X i =1 k f i k q ! /q ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X i =1 | f i | q ! /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , f i ∈ L . Any quasi-Banach lattice is trivially ∞ -concave. Theorem 6.3.
Let X be a quasi-Banach space with a normalized un-conditional basis X = ( x n ) ∞ n =1 . Suppose that X induces a q -concavelattice structure on X for some < q ≤ ∞ . Let w = ( w n ) ∞ n =1 be aweight with s m := m X n =1 w n ≤ ϕ l [ X ]( m ) , m ∈ N . Then X dominates the unit vector system of d ,q ( w ) , i.e., X continu-ously embeds into d ,q ( w ) via X .Proof. If q = ∞ the result is known (see [4, Lemma 6.1]). Suppose that q < ∞ . Put w ′ = ( w ′ n ) ∞ n =1 , where w ′ n = s qn − s qn − with the conventionthat s = 0. Let ( a n ) ∞ n =1 ∈ c such that ( | a n | ) ∞ n =1 is non-increasing.Put t = | a | and for each k ∈ N consider the set J k = { n ∈ N : t − k < | a n | ≤ t − k +1 } . Notice that ( J k ) ∞ k =1 is a partition of { n ∈ N : a n = 0 } . Set m = 0 andfor j ∈ N put m j = P jk =1 | J k | , so that J j = { n ∈ N : m j − + 1 ≤ n ≤ m j } for all j ∈ N . Define f k = 2 k X n ∈ J k a n x n . By Abel’s summation formula, f := ∞ X n =1 a n x n = ∞ X k =1 − k f k = 12 ∞ X k =1 − j j X k =1 f k . Therefore, if C is the q -concavity constant of X and K is its uncondi-tionality basis constant, k f k q ≥ t q q C q K q ∞ X j =1 − jq s qm j = t q q C q K q ∞ X j =1 − jq j X k =1 X n = J k w ′ n = (2 q − t q C q K q ∞ X k =1 − kq X n = J k w ′ n = (2 q − q C q K q ∞ X n =1 | a n | q w ′ n . Using (6.3), and taking into account that the concavity constants andthe unconditionality constants of ( x π ( n ) ) ∞ n =1 are still C and K for anypermutation π of N , we are done. (cid:3) We also need the dual property of URP. A sequence (Φ( m )) ∞ m =1 ofpositive scalars is said to have the lower regularity property (LRP forshort) if there is r ∈ N such thatΦ( rn ) ≥ n ) n ∈ N . (Φ( m )) ∞ m =1 has the LRP if and only if ( m/ Φ( m )) ∞ m =1 has the URP.Hence a dual inequality of (6.4) holds, i.e., for any sequence (Φ( m )) ∞ m =1 with the LRP there is a constant C such that m X n =1 Φ( n ) n ≤ C Φ( m ) , m ∈ N . Lemma 6.4.
Suppose that a sequence space L on a set J is q -concavefor some < q < ∞ . The ϕ l [ L ] has the LRP.Proof. Let r , m ∈ N , and A ⊆ J with | A | = rm . Pick a partition( A j ) rj =1 of A with | A j | = m for all j = 1, . . . r . If C is the q -concavityconstant of L , k A k L ≥ C q X j =1 k A j k q L ! /r ≥ r /q C ϕ l [ L ]( m ) . Hence, if we pick r ≥ (2 C ) q we get ϕ l [ L ]( rm ) ≥ ϕ l [ L ]( m ). (cid:3) Even without having any information on the concavity of the space X , Proposition 6.5 below provides an improvement of [2, Proposi-tion 5.6]. In addition to that, it shows that imposing some nontrivialconcavity to the lattice structure allows to weaken the assumption onthe lower democracy function. Proposition 6.5.
Let X be a quasi-Banach space with a normalizedunconditional basis X . Suppose that X induces a q -concave lattice structure, ≤ q ≤ ∞ . Denote by q ′ the conjugate exponent of q ,and put s m = ϕ l [ X ]( m ) for all m ∈ N . Suppose that either q = 1 and lim m s m m = ∞ , or q > and ∞ X m =1 m q ′ − s q ′ m < ∞ . Then X is strongly absolute.Proof. If 1 ≤ q < ∞ , applying Lemma 6.4 gives a constant C such that m X n =1 s n n ≤ Cs m , m ∈ N . Set w = s n /n ) ∞ n =1 if 1 < q < ∞ , and let w be the weight whoseprimitive weight is ( s m ) ∞ m =1 if q ∈ { , ∞} . By Proposition 6.2, the unitvector system of d ,q ( w ) is strongly absolute. Then, the result followsfrom combining Theorem 6.3 with Lemma 3.19. (cid:3) Example . Let w = ( w n ) ∞ n =1 be a weight whose primitive weight( s n ) ∞ n =1 is doubling.(i) If P ∞ n =1 /s n < ∞ , the spaces ℓ p ( d , ∞ ( w )) = ( d , ∞ ( w ) ⊕ · · · ⊕ d , ∞ ( w ) ⊕ · · · ) ℓ p ,d , ∞ ( w )( ℓ p ) = ( ℓ p ⊕ · · · ⊕ ℓ p ⊕ · · · ) d , ∞ ( w ) have a (UTAP) unconditional basis for all 0 < p < < q < ∞ and denote by q ′ its conjugate exponent. Supposethat P ∞ n =1 w − q ′ +1 n s − n < ∞ . Then the spaces ℓ p ( d ,q ( w ) = ( d ,q ( w ) ⊕ · · · ⊕ d ,q ( w ) ⊕ · · · ) ℓ p ,d ,q ( w )( ℓ p ) = ( ℓ p ⊕ · · · ⊕ ℓ p ⊕ · · · ) d ,q ( w ) have a (UTAP) unconditional basis for all 0 < p < m s n /n >
0, then the space ℓ p ( d ,q ( w )) = ( d ,q ( w ) ⊕ · · · ⊕ d ,q ( w ) ⊕ · · · ) ℓ p has a (UTAP) unconditional basis for all 0 < p < < q < d ,q ( w ) is, for a different weight w ′ , the classical space d ( q, w ′ ), considered in [9] (see 6.1).(iv) Let 0 < q ≤ n s n /n = ∞ . Then the spaces ℓ p ( d ,q ( w )) and d ,q ( w )( ℓ p ) have a (UTAP) unconditional basis forall 0 < p ≤ Orlicz sequence spaces. A normalized Orlicz function is a right-continuous increasing function F : [0 , ∞ ) → [0 , ∞ ) such F (0) = 0 and F (1) = 1. The topological vector space built from the modular( a n ) ∞ n =1 ∞ X n =1 F ( | a n | )is the Orlicz sequence space usually denoted by ℓ F . The space ℓ F islocally bounded if and only if If there is p >
Let F be a normalized Orlicz function and set G ( t ) = t/F ( t ) , < t < ∞ . Suppose that G doubling near the origin, essentiallyincreasing, and satisfies lim t → + G ( t ) = 0 . Then F is doubling near theorigin, the Orlicz sequence space ℓ F is minimal, and the canonical basisis strongly absolute.Proof. Let C ∈ [1 , ∞ ) be such that G ( s ) ≤ CG ( t ) for all 0 < s ≤ t ≤ F ( t ) = tG ( t ) ≤ CtG ( s ) = Cts F ( s ) , < s ≤ t ≤ ,F is doubling near the origin.Fix R < ∞ and pick δ > G ( t ) ≤ / ( RC ) for every0 < t ≤ δ . Given f = ( a n ) ∞ n =1 ∈ ℓ F , set u = k f k ∞ and v = k f k ℓ F .Then w := ∞ X n =1 | a n | = ∞ X n =1 vF (cid:18) | a n | v (cid:19) G (cid:18) | a n | v (cid:19) ≤ CvG (cid:16) uv (cid:17) ∞ X n =1 F (cid:18) | a n | v (cid:19) ≤ CvG (cid:16) uv (cid:17) . In the case when u/v ≤ δ we have w ≤ v/R . Otherwise, w ≤ CvG (cid:16) uv (cid:17) = CuF ( u/v ) ≤ CuF ( δ ) . Hence, ℓ F is strongly absolute with function α given by α ( R ) = CF (inf { t : RCG ( t ) < } ) , < R < ∞ . (cid:3) Let us mention that Kalton [21] (implicitly) proved that if F and itsdual function G are doubling near the origin, G is bounded near theorigin, and lim ε → + inf G ( t ) = t/F ( t ). Assume that F and G are doubling near the origin,that G is essentially increasing, and that lim t → + G ( t ) = 0. Then thefollowing spaces have a (UTAP) unconditional basis: (i) ℓ ( ℓ F ) = ( ℓ F ⊕ ℓ F ⊕ · · · ⊕ ℓ F ⊕ · · · ) ;(ii) ( L ∞ n =1 ℓ nq ) ℓ F = ( ℓ q ⊕ ℓ q ⊕ · · · ⊕ ℓ nq ⊕ · · · ) ℓ F for all 0 < q ≤ ℓ F ( c ) = ( c ⊕ c ⊕ · · · ⊕ c ⊕ · · · ) ℓ F ;(iv) ℓ F ( d , ∞ ( w )) = ( d , ∞ ( w ) ⊕ d , ∞ ( w ) ⊕ · · · d , ∞ ( w ) . . . ) ℓ F and d , ∞ ( w )( ℓ F ) = ( ℓ F ⊕ ℓ F ⊕ · · · ℓ F . . . ) d , ∞ ( w ) , where w is as in Ex-ample 6.6 (i).(v) ℓ F ( d ,q ( w ) = ( d ,q ( w ) ⊕ d ,q ( w ) ⊕ · · · d ,q ( w ) . . . ) ℓ F and d ,q ( w )( ℓ F ) = ( ℓ F ⊕ ℓ F ⊕ · · · ℓ F . . . ) d ,q ( w ) , where w and q are as inExample 6.6 (ii) and (iv).For instance, the uniqueness of unconditional basis in ℓ ( ℓ F ) is anapplication of Theorem 5.3. To see (ii) when 0 < q <
1, we just need toapply Theorem 3.18 since condition (c) in Lemma 3.4 is fulfiled. Thenby Proposition 3.20, the canonical basis of ℓ F ( ℓ q ) is strongly absoluteand equivalent to its square. To show the case when q = 1 in part (ii)and part (iii) however, we need to appeal to Theorem 4.2 and take intoaccount that the unit vector basis of ℓ and c is perfectly homogeneous,hence uniformly universal for well complemented block basic sequenceswith function C
1. The verification of the corresponding hypothesesleading to the uniqueness property in the remaining cases is totallystraightforward, and so we leave it for the reader.6.3.
Bourgin-Nakano spaces. 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 the modular m ( p n ) : F N → [0 , ∞ ) , ( a n ) n ∈ N X n ∈ N | a n | p n . 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 minimal sequence space. We have ℓ ( p n ) = h ( p n ) if and only ifsup n p n < ∞ .The unit vector system ( e n ) ∞ n =1 of ℓ ( p n ) is a 1-unconditional basiswhich is universal for well complemented block basic sequences ([2,Proposition 4.7]), and which is strongly absolute if and only if q :=lim sup p n <
1. Indeed, this condition implies that the space embedsnaturally into ℓ q and so we can apply Lemma 3.19. Moreover, theBanach envelope of ℓ ( p n ) is anti-Euclidean if and only if lim sup p n ≤ C C . Example . The following spaces have a (UTAP) unconditional basis:(i) ℓ ( p n )( ℓ ) = { ( z n ) ∞ n =1 : z n ∈ ℓ and P ∞ n =1 k z n k p n < ∞} , wherelim sup p n < ℓ F ( ℓ ( p n )) = ( ℓ ( p n ) ⊕ ℓ ( p n ) ⊕ · · · ⊕ ℓ ( p n ) ⊕ · · · ) ℓ F , where F is anOrlicz function as in Example 6.8 and lim sup p n ≤ ℓ ( ℓ ( p n )) = ( ℓ ( p n ) ⊕ ℓ ( p n ) ⊕ · · · ⊕ ℓ ( p n ) ⊕ · · · ) ℓ q , where lim p n < ℓ is equivalent to its square (condition (iv) (a)). In thesecond example we use also Theorem 4.2, but now we employ condition(iv) (b) since, while ℓ ( p n ) need not be lattice isomorphic to its square, ℓ F is. The last case is just a direct application of Theorem 5.3, since thehypothesis ensures that the canonical basis of ℓ ( p n ) is strongly absolute.Note that in the cases (ii) and (iii), the uniqueness of unconditionalbasis in the direct sum is obtained without knowing whether the space ℓ ( p n )) has a unique unconditional basis or not!6.4. Hardy spaces.
Because of their importance in Analysis, we sin-gle out as well some examples involving Hardy spaces. For the con-venience of the reader we will next state a few known facts about thespaces H p ( T d ) that we will need in order to apply Theorems 4.2 and5.3.The first unconditional bases in H p ( T ) for 0 < p < H = ( x n ) ∞ n =0 is such a normalized basis then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =0 a n x n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H p ( T ) ≈ Z ∞ X n =0 | a n | h n ! p/ /p , ( a n ) ∞ n =1 ∈ c , (6.6)where ( h n ) ∞ n =0 is the classical Haar system on [0 ,
1] normalized with re-spect to the norm in L p ([0 , H p ( T d ) for d ∈ N which satisfy an equiva-lence analogous to (6.6). Using those tensored bases, Kalton et al. [24]showed that the spaces H p ( T d ) and H p ( T d ′ ) with 0 < p < d , d ′ ∈ N , are isomorphic if and only if d = d ′ . Then it was proved in[36] that all the spaces H p ( T d ) for 0 < p < d ∈ N have a UTAPunconditional basis.The canonical basis H of the Hardy spaces H p ( T d ), 0 < p < p -convex lattice structure and satisfies the estimate m /p ≈ ϕ lm [ H , H p ( T d )] , m ∈ N . Hence, Proposition 6.5 implies that H is strongly absolute This way wecan use Hardy spaces (or more generally subspaces of Hardy spaces gen-erated by subbases of the Haar system) to build examples of spaces witha (UTAP) unconditional basis. Given a (finite or infinite) nonemptysubset n ⊆ N , H n p ( T ) denotes the subspace of H p ( T ) generated byHaar functions belonging to layers in n . Example . The following spaces have a (UTAP) unconditional ba-sis:(i) The space H p ( T , d ,q ( w )) = d ,q ⊕ d ,q ( w ) ⊕ · · · ⊕ d ,q ( w ) ⊕ · · · ) H p ,consisting of all sequences ( z n ) ∞ n =1 such that z n ∈ d ,q ( w ) for all n ∈ N and Z ∞ X n =0 k z n k h n ( t ) ! p/ dt /p < ∞ , where p <
1, 0 < q < s m ) ∞ m =1 of w satisfies inf m s m /m > ℓ F ( H p ( T )) = ( H p ( T ) ⊕ H p ( T ) ⊕ · · · ⊕ H p ( T ) ⊕ · · · ) ℓ F , where F isas in Proposition 6.7.(iii) ( L ∞ k =1 H n k p ( T )) ℓ = ( H n p ( T ) ⊕ H n p ( T ) ⊕ · · · ⊕ H n k p ( T ) ⊕ · · · ) ℓ ,where 0 < p < n k ) ∞ k =1 is an increasing sequence of subsetsof N .Note that in the last example, since there are sets of layers n ⊆ N for which H n p ( T ) is not isomorphic to its square (see [36]), we mustuse condition (b) (ii) in order for all the hypotheses of Theorem 5.3 tobe satisfied. As a matter of fact, it is unknown whether H n p ( T ) has aUTAP unconditional basis in the case when it is not isomorphic to itssquare (see Theorem 3.18).6.5. Tsirelson’s space.
Casazza and Kalton established in [15] 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. ([13]), who had proved the uniqueness of unconditionalbasis up to permutation of the 2-convexifyed Tsirelson’s space T (2) .Unlike T (2) , which is “highly” Euclidean, the space T is anti-Euclidean.To see the latter requires the notion of dominance, introduced in [15].Let X = ( x n ) ∞ n =1 be a (normalized) unconditional basis of a quasi-Banach space X . Given f , g ∈ X , we write f ≺ g if m < n for all m ∈ supp( f ) and n ∈ supp( g ). Given D ≥
1, the basis X is said to be left (resp. right) D -dominant if whenever ( f i ) ni =1 and ( g i ) ni =1 are disjointlysupported families with f i ≺ g i (resp. g i ≺ f i ) and k f i k ≤ k g i k forall i = 1, . . . , n , then k P ni =1 f i k ≤ D k P ni =1 g i k . As is customary, ifthe constant D is irrelevant, we just drop it from the notation. If X is a Banach space with a left (resp. right) dominant semi-normalizedunconditional basis X there is a unique r = r ( X ) ∈ [1 , ∞ ] such that ℓ r is finitely block representable in X . In the case when r ( X ) ∈ { , ∞} , X is anti-Euclidean (see [15, Proposition 5.3]).The canonical basis of the Tsirelson space T is (normalized, 1-unconditional and) right 16-dominant (see [15, Proposition 5.12]) with r ( T ) = 1. In turn, by [15, Lemma 5.1], the canonical basis of theoriginal Tsirelson’s space T ∗ is left-dominant.Moreover, by [15, Proposition 5.5] and [17, page 14], the canonicalbases of T and T ∗ (as well as each of their subases) are equivalent totheir square. In our language, [15, Theorem 5.6] says that every left(resp. right) dominant unconditional basis is universal for well comple-mented block basic sequences. Combining the arguments used in itsproof with Lemma 3.14 yields the following quantitative result. Theorem 6.11.
Let X = ( x k ) ∞ k =1 be a left (or right) D -dominant nor-malized K -unconditional basis of a quasi-Banach space X with modulusof concavity at most κ . Then X is uniformly universal for well com-plemented block basic sequences with function depending on D , K and κ .Proof. Let us do the right-dominant case only as the left-dominant caseis similar. For that, we first show that there are constants D and D depending only on D , κ and K such that any semi-normalized disjointlysupported basic sequence U = ( u m ) m ∈M D -dominates ( a x k m ) m ∈M and it is D -dominated by ( b x j m ) m ∈M , where a = inf m k u m k , b =sup m k u m k , j m = min(supp( y m )), and k m = max(supp( y m )).Indeed, if A m = supp( u m ) \ { j m } , and ( a m ) m ∈M ∈ c ( M ), (cid:13)(cid:13)(cid:13) X m ∈M a m u m (cid:13)(cid:13)(cid:13) ≤ κ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈M a m S A m ( u m ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈M a m x ∗ j m ( u m ) x j m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)! ≤ κD (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈M a m k S A m ( u m ) k x j m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈M a m x ∗ j m ( u m ) x j m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)! ≤ κKDb (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈M a m x j m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . In turn, if F m = supp( u m ) \ { k m } , there are ( λ m ) m ∈M and ( γ m ) m ∈M such that a = κ ( λ m + γ m ), 0 ≤ λ m ≤ k S F m ( u m ) k , and 0 ≤ γ m ≤ | x ∗ k m ( u m ) | for all m ∈ M . Hence, (cid:13)(cid:13)(cid:13) X m ∈M a m u m (cid:13)(cid:13)(cid:13) ≥ K max ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈M a m λ m S F m ( u m ) k S F m ( u m ) k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈M a m γ m x k m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)) ≥ KD max ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈M a m λ m x k m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈M a m γ m x k m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)) ≥ a κ KD (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈M a m x k m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Pick 0 < λ <
1. Let Y = ( y m ) m ∈M be a well complementednormalized basic sequence with good C -projecting functionals Y =( y ∗ m ) m ∈M . For each m ∈ M there is k m ∈ supp( y m ) such that, if A lm = supp( y m ) ∩ [1 , k m ] and A rm = supp( y m ) ∩ [ k m , ∞ ), | y ∗ m ( S A lm ( y m )) | ≥ λ, | y ∗ m ( S A rm ( y m )) | ≥ − λ. By Lemma 3.14, Y r := ( S A rm ( y m )) m ∈M ( CK/ (1 − λ ))-dominates Y ,and Y ( CK )-dominates Y l := ( S A lm ( y m )) m ∈M . Moreover Y l ( CK/λ )-dominates Y , whence k S A lm ( y m ) k ≥ λCK , m ∈ M . Therefore, Y l ( CKD /λ )-dominates ( x k m ) m ∈M . Since k S A rm ( y m ) k ≤ K for all m ∈ M , ( x k m ) m ∈M ( KD )-dominates Y r . Summing up,choosing λ = 1 / (1 + κCK ) we infer that X is uniformly universal forwell complemented basic sequences with function C κCK D (1 + κC ) . (cid:3) Finally, since they are locally convex, both T and T ∗ are trivially L -convex lattices.Combining the above background information with our main resultswe bring out a couple of examples: Example . For 0 < p < F an Orlicz function as in Proposi-tion 6.7, the spaces(i) ℓ p ( T ) = ( T ⊕ T ⊕ · · · ⊕ T ⊕ · · · ) ℓ p and(ii) ℓ F ( T ∗ ) = ( T ∗ ⊕ T ∗ ⊕ · · · ⊕ T ∗ ⊕ · · · ) ℓ F have a (UTAP) unconditional basis. Mixed-norm Lebesgue sequence spaces.
We close with ap-plications to finite and infinite direct sums of mixed-norm Lebesguesequence spaces.
Example . Suppose ( p j ) nj =1 is sequence of indexes in (0 ,
1] with p j = 1 for at most one j . We consider the space X = ℓ p ( ℓ p ( · · · ℓ p i ( · · · ( ℓ p n )))of recursive direct sums of a finite number of (possibly repeated) se-quence spaces ℓ p j . X is a p -Banach space for p = min j p j , and itscanonical basis X is unconditional and equivalent to its square. More-over, X induces on X a p -convex lattice structure, and it dominatesthe unit vector system of ℓ q , where q := max j p j . Thus, in the casewhen q <
1, Lemma 3.19 implies that X is also strongly absolute.Therefore, by Theorem 3.18, X has a (UTAP) unconditional basis. Ifwe let one (and only one) of the indexes p j be 1, we need to distin-guish two cases. Suppose first that p = 1 and 0 < p j < 1] with s :=inf n p n > 0, and let q ∈ (0 , X = ∞ M n =1 ℓ p n ! ℓ q = ( ℓ p ⊕ ℓ p ⊕ · · · ⊕ ℓ p n ⊕ · · · ) ℓ q . Note that since ℓ p ( ℓ p ) = ℓ p isometrically for all p > 0, there is no realrestriction in assuming that the indices p j are not repeated.The unit vector system E p of ℓ p is 2 /p equivalent to its square. More-over, E p is perfectly homogeneous, thus uniformly universal for wellcomplemented block basic sequences with function C 1. Finally, E p is 1-unconditional and, if we consider on ℓ p the lattice structureinduced by E p , M r ( ℓ p ) = 1 for all r ≤ p . Hence, in the case when q < 1, the uniqueness of unconditional basis of X is an application ofTheorem 4.2, where the hypothesis (iv) is fulfilled with condition (a).Suppose now that q = 1 and t := sup n p n < 1. The important detailhere is that all the canonical bases of ℓ p n are strongly absolute with thesame function α . In fact, by [5, Lemma 3.2], we can choose α ( R ) = ( R t/ (1 − t ) if R ≥ ,R s/ (1 − s ) if R ≤ . Hence, applying Theorem 5.3 gives that X has a (UTAP) unconditionalbasis. Remark . In Example 6.15, We do not known whether X has aUTAP unconditional basis in the case when q = 1 and lim n p n = 1. References [1] F. Albiac and J. L. Ansorena, Projections and unconditional bases in directsums of ℓ p spaces, < p ≤ ∞ , arXiv e-prints (2019), available at .accepted for publication in Mathematische Nachrichten.[2] , On the permutative equivalence of squares of unconditional bases ,arXiv e-prints (2020), available at .[3] F. Albiac, J. L. Ansorena, P. M. Bern´a, and P. Wojtaszczyk, Greedy approxi-mation for biorthogonal systems in quasi-banach spaces , arXiv (2019), availableat .[4] F. Albiac, J. L. Ansorena, S. J. Dilworth, and D. Kutzarova, Banach spaceswith a unique greedy basis , J. Approx. Theory (2016), 80–102.[5] F. Albiac, J. L. Ansorena, and P. Wojtaszczyk, Quasi-greedy bases in ℓ p (
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