The structure of symmetric basic sequences with applications to a class of Orlicz sequence spaces

J. Math. Anal. Appl. 426 (2015) 380–391

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

The structure of symmetric basic sequences with applications to a class of Orlicz sequence spaces Carlos E. Finol a , Marcos J. González b,∗ a

Escuela de Matemáticas, Universidad Central de Venezuela, P.O. Box 48059, Caracas, 1041-A, Venezuela b Departamento de Matemáticas, Universidad Simón Bolívar, Apartado 89000, Caracas, 1080-A, Venezuela

a r t i c l e

i n f o

Article history: Received 22 November 2013 Available online 30 October 2014 Submitted by Richard M. Aron Keywords: Banach spaces: symmetric bases Banach sequence spaces: Orlicz, Lorentz Functions: geometrically convex, super-multiplicative, slowly varying

a b s t r a c t We prove the following general result: Let (xn ) be a boundedly complete symmetric basis for a Banach space X. Then, for every symmetric basic sequence in X, we have the following alternatives: (a) it is equivalent to a basic sequence generated by a vector with respect to (xn ), or (b) it dominates a normalized block basis of (xn ) having coefficients tending to zero. This is an extension of a similar result obtained in 1973 by Altshuler, Casazza and Lin [1] for Lorentz sequence spaces. As an application, we obtain that, if M is a geometrically convex Orlicz function, then every symmetric basic sequence in the Orlicz sequence space M has the property (a) above, or it is equivalent to the standard basis of an p -space. © 2014 Published by Elsevier Inc.

1. Introduction This paper deals with the structure of Banach spaces with a symmetric boundedly complete Schauder basis. We use standard notations and, for undefined notions below, see the next section. If X is a Banach space with a basis (xn ), then (x∗n ) denotes the sequence of functionals biorthogonal to the basis. To shorten the text, the symbol (epn ) denotes the standard unit vector basis of the Banach space p, where 1 ≤ p < ∞. In 1973, Altshuler, Casazza and Lin [1] studied the structure of symmetric basic sequences in Lorentz sequence spaces. A useful tool in their studies has turned out to be the following notion. ∞ Definition 1.1. Let (xn )∞ n=1 be a symmetric basis of a Banach space X, and let α = n=1 an xn be a non-zero vector of X. A basic sequence (un )∞ in X is generated by the vector α, with respect to (xn )∞ n=1 n=1 , if there exists a bijection φ: N × N → N such that, for each n ∈ N, the sequence fn (m) = φ(m, n) is increasing and ∞ un = m=1 am xφ(m,n) . * Corresponding author. E-mail addresses: carlos.fi[email protected] (C.E. Finol), [email protected] (M.J. González). http://dx.doi.org/10.1016/j.jmaa.2014.10.073 0022-247X/© 2014 Published by Elsevier Inc.

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This notion has been applied to obtain a complete classification of symmetric basic sequences in concrete spaces (cf., [1, Theorem 6], [12, Example 3.b.10], [6, Theorem X.b.10]). The following result (cf., [1, Theorem 3]) is a key property for this classification in the Lorentz sequence spaces and one of our main goals is its extension – in our Theorem 1.3 – to a class of Orlicz spaces. Proposition 1.1. (See Altshuler–Casazza–Lin [1].) Let a = (an ) ∈ c0  1 be a non-increasing positive sequence, let 1 ≤ p < ∞, and let (ea,p n ) denote the standard unit vector basis of the Lorentz sequence space d(a, p). Then, every symmetric basic sequence in d(a, p) is equivalent to either the basis (epn ) or a basic sequence generated by one vector with respect to (ea,p n ). We would like to add that both, the above proposition and Theorem 1.3, are particular cases of our Theorem 1.2 presented below. Another useful notion in the study of the geometry of Banach spaces having symmetric bases, which goes back to Lindberg [11], is the notion of normalized block bases having coefficients tending to zero. Definition 1.2. Let (xn ) be a basis of a Banach space X, and let (yn ) be a normalized block basic sequence of (xn ). We say that (yn ) is vanishing with respect to (xn ) if the sequence consisting of all the non-zero coefficients of the vectors in (yn ), with respect to (xn ), tends to zero. In other words, a vanishing block of (xn ) is a normalized block basis (yn ) of (xn ) such that limn→∞ (maxj∈supp yn {|x∗j (yn )|}) = 0, where supp y denotes the support of y with respect to (xn ). For example, the following is known: Let (en ) be the sequence of unit vectors in a symmetric sequence space E. Let (yn ) be a vanishing block basis of (en ). Then: (A) if E is an Orlicz sequence space M , then (yn ) has a subsequence equivalent to the unit vector basis of hN , for some N ∈ CM . This is used implicitly in the proof of [12, Proposition 4.b.10] and has been proved for the case of Orlicz–Lorentz sequence spaces in [10] (see Proposition 2.11 below). (B) If E is a Lorentz sequence space d(a, p), Altshuler, Casazza and Lin (cf., [12, Proposition 4.e.3]) have proved that (yn ) has a subsequence equivalent to (epn ). (C) The case when E is an Orlicz–Lorentz [resp., the predual to a Lorentz] sequence space is studied in [10] (resp., [5]). Our main result is the following selection principle: Theorem 1.2. Let (xn ) be a boundedly complete symmetric basis for a Banach space X. Then, for every symmetric basic sequence (yn ) in X, one of the following holds true: (a) the basic sequence (yn ) is equivalent to a basic sequence generated by a vector with respect to (xn), or (b) the basic sequence (yn ) dominates a vanishing block basis of (xn ). This theorem applies whenever each vanishing block basis has ‘increasingly dominant subsequences’. For example, by the statement (B) mentioned above, we see that Proposition 1.1 can be seen as a particular case of Theorem 1.2. Given A > 0, we set IA = (0, A] or [A, ∞). A function f : IA → R+ is geometrically convex, if the inequality f (xα y β ) ≤ f (x)α f (y)β , holds for every α, β ≥ 0, α + β = 1, x, y ∈ IA . If the reverse inequality holds, then f is geometrically concave. The multiplicative properties of these functions are studied in [7]. Motivated by [1, Theorem 3], for the collection of symmetric basic sequences of a given Orlicz sequence space M , corresponding to a geometrically convex Orlicz function M , we have obtained the following decomposition, up to equivalence, of the set of symmetric basic sequence in M .

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Theorem 1.3. Let M be an Orlicz function which is geometrically convex on (0, A] for some A > 0. Let (xn ) M (t) be a normalized basic sequence in the Orlicz sequence space M . Then, the limit p := limt→0+ loglog exists t M and is finite, the unit vectors form a symmetric basis (en ) for M , and (xn ) is (a) equivalent to the p -basis (epn ), or (b) is equivalent to a basic sequence generated by a vector in M with respect to the basis (eM n ). 2. Preliminaries 2.1. Geometrically convex functions Let I ⊂ [0, ∞) be such that st ∈ I, if s, t ∈ I, e.g., the intervals (0, A], [1/A, ∞), for 0 < A ≤ 1. A function f : I → [0, ∞) is sub-multiplicative if, for every s, t ∈ I, f (st) ≤ f (s)f (t). If the reverse inequality holds, then f is super-multiplicative. The following result is a particular case of [7, Theorem 1]: Proposition 2.1. Let f : IA → R+ be a continuous function. The following conditions are equivalent: (i) The function is geometrically convex [resp., concave] on IA . (ii) For every a ∈ IA , the function fa (t) := ff(at) (a) is super-[resp., sub-] multiplicative. (iii) For every t ∈ IA , the function f {t} (s) := [resp., non-decreasing].

f (ts) f (s)

is non-increasing

2.2. Bases and basic sequences We refer to the monographs [12] and [15] for terminology and notations. Recall that a basis (xn ) is boundedly complete if, for every scalar sequence (αn ), the condition m ∞ supm≥1  n=1 αn xn  < ∞ implies that the series n=1 αn xn converges in X. We will use the following result (cf., [12, Theorem 1.c.10]): for a Banach space X, having an unconditional basis (xn ), the space c0 embeds isomorphically into X if and only if the basis (xn ) is not boundedly complete. Given two basic sequences (xn ) and (yn ) and K > 0, (xn ) K-dominates (yn ) (or (yn ) is 1/K-dominated m m by (xn )) if for every integer m ≥ 1 and every scalar sequence (αn ),  n=1 αn xn  ≤ K n=1 αn yn . The basic sequences (xn ) and (yn ) are K-equivalent if there are constants K1 , K2 > 0 such that K1 /K2 ≤ K, and (xn ) simultaneously K1 -dominates and is K2 -dominated by (yn ). For future reference, we state the following criterium for equivalence of basic sequences (cf., [12, Proposition 1.a.9]): Proposition 2.2. Let (xn ) be a basic sequence in a Banach space X, and let (yn ) be a sequence in X such ∞ 1+λ that n=1 xn − yn  x∗n  = λ < 1. Then, (yn ) is a basic sequence which is 1−λ -equivalent to (xn )∞ n=1 . We will use the following basic fact taken from [2] (cf., [12, Proposition 1.a.12]): Proposition 2.3 (Bessaga–Pełczyński). Let (xn ) be a normalized basis of a Banach space X, and let (yn ) be a sequence in X such that, for every n ≥ 1, we have yn  ≥ δ > 0 and limm→∞ x∗m (yn ) = 0. Then, for every ε > 0, there exists a subsequence of (yn ) which is (1 + ε)-equivalent to a normalized block basis of (xn ). A basis (xn ) is unconditional if each permutation of (xn ) is also a basis; in this case, the suppression  unconditional norm, defined for all x = n αn xn , by the expression   m     xu = sup sup  θn αn xn ,   m≥1 θ∞ ≤1 n=1

is an equivalent norm in X and ubc(xn ) denotes the least constant K > 0 so that xu ≤ Kx.

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Another basic fact developed in [3, 1.1] is the following (cf., [1, Proposition 1]). Proposition 2.4 (Bessaga–Pełczyński). Let (yn ) be an unconditional basis for a Banach space X. Suppose there exists a functional x∗ ∈ X ∗ such that 1/C := lim supn→∞ |x∗ (yn )| > 0. Set D := Cx∗ ubc(yn ), where ubc(yn ) is the suppression unconditional constant of (yn ). Then, there exists a subsequence of (yn ) which is D-equivalent to the unit vector basis (e1n ) of 1 . In particular, every symmetric basic sequence is weakly null or equivalent to the basis (e1n ). A basis (xn ) is symmetric if it is unconditional and equivalent to each of its permutation. Every symmetric basis is subsymmetric, i.e., it is equivalent to each of its subsequences (cf., [12, Proposition 3.a.3]) and semi-normalized or bounded, i.e., 0 < inf n {xn } ≤ supn {xn } < ∞ (cf., [15, Proposition 21.4., p. 569]).  The symmetric norm, defined for all x = n αn xn , by the expression   m     |x| = sup sup sup  θn αn xπ(n) ,  π∈Π θ∞ ≤1 m≥1 n=1

is an equivalent norm in X. The symmetric basis constant sbc(xn ) is the least constant K > 0 so that |x| ≤ Kx. If sbc(xn ) = 1, we say that (xn ) is 1-symmetric (cf., [15, Definition 22.1, p. 574], [15, Definition 22.2, p. 582]). We will use [5, Proposition 1], which describes a kind of (semi-normalized) block basic sequences which are equivalent to (un ). Proposition 2.5 (Altshuler). Let (xn ) be a 1-symmetric basis of a Banach space X. Then, for every basic ∞ sequence (un ) in X, generated by the vector α = k=1 ak xk with respect to (xn ), and for every ε > 0, there is an increasing sequence of integers (pn )∞ n=1 , with p1 = 0, such that (un ) is (1 + ε)-equivalent to the block pn+1 basic sequence (yn ) of the form yn = k=pn +1 ak−pn xk , for every n ≥ 1. We will also use [1, Proposition 3], which asserts that the normalized block bases of a symmetric basis, with blocks having uniformly bounded sizes, are equivalent to the basis. pn+1 αk xk , n = 1, 2, . . . Proposition 2.6. Let (xn ) be a subsymmetric basis of a Banach space X. If yn = k=p n +1 is a normalized block basis of (xn ) such that supn≥1 {pn+1 − pn } < ∞, then (yn ) is equivalent to (xn ). 2.3. Orlicz sequence spaces Our terminology and notations are standard and follow [11], the monograph by [12] and the more recent approach of Kamińska and Raynaud [10]. An Orlicz function M is a non-negative, non-decreasing, convex function defined on [0, ∞), with M (0) = 0. If M (t) > 0 for all t > 0 then M is called non-degenerate. Let RN denote the space of all  scalar sequences and let M : RN → [0, ∞] be defined by the formula M (a) = M (|an |), where a ∈ RN . The Orlicz sequence space M is defined as the linear set   M := a ∈ RN : M (a/λ) < ∞ for some λ > 0 , and equipped with the norm aM := inf{λ > 0: M (a/λ) ≤ 1} it becomes a Banach space. By hM we denote the closed subspace of M defined as   hM := a ∈ RN : M (a/λ) < ∞ for all λ > 0 ,

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where the unit vectors form a symmetric basis of hM (cf., [12, Proposition 4.a.2, p. 138]). An Orlicz function M fulfills the Δ2 -condition at 0 if M is non-degenerate and there exist constants K, t0 > 0 with 0 < M (2t) ≤ KM (t), for every t ∈ [0, t0 ]. The following holds true (cf., [12, Proposition 4.a.4]): Proposition 2.7. Let M be an Orlicz function. Then the following five conditions are equivalent: (i) The function M fulfills the Δ2 -condition at zero; (ii) M = hM ; (iii) The unit vectors form a boundedly complete symmetric basis of hM ; (iv) The space M is separable. Let M , N be two Orlicz functions. We say that M dominates N at 0 if there exist constants a, t0 > 0, such that M (at) ≥ N (t) for every t ∈ [0, t0 ]. The functions are equivalent at 0 if M dominates N and N dominates M . We will use the following basic fact (cf., [13, Theorem 3.4, pp. 18–19]): Proposition 2.8. Let M , N be two Orlicz functions. Then, the following conditions are equivalent: (i) M ⊂ N ; (ii) the unit vector basis of hM dominates the unit vector basis of hN ; (iii) M dominates N . An Orlicz function M can be regarded as an element of the cube [0, ∞][0,∞] , where [0, ∞] is the one-point compactification of [0, ∞). By the Tychonoff theorem, [0, ∞][0,∞] is a compact Hausdorff space under the product topology, which coincides with that of pointwise convergence. Given a non-degenerate Orlicz function M and λ > 0, we denote by Mλ the Orlicz function Mλ (t) := M (λt) [0,∞] 0 0 , for 0 < Λ ≤ ∞: CM,Λ := conv EM,Λ where M (λ) , for all t ∈ [0, ∞). Consider the subsets of [0, ∞] 0 EM,Λ := {Mλ : 0 < λ < Λ}, and set 0 EM,Λ := EM,Λ = {Mλ ; 0 < λ < Λ}, 0 CM,Λ := CM,Λ = conv EM,Λ ,

CM =

EM := 



EM,Λ

(1)

Λ>0

CM,Λ .

(2)

Λ>0

Here the closure is taken with respect to the product topology. Then, CM,Λ and EM,Λ are non-empty compact subsets of [0, ∞][0,∞] and, consequently, the intersections CM and EM are non-empty and compact as well. A proof of the following basic fact can be found in [8, Lemma 2.4]. Proposition 2.9. Let M be a non-degenerate Orlicz function. (λn t) (a) If N ∈ EM,λ then, for every fixed t > 0, there exists a sequence {λn } ⊂ (0, λ) such that limn→∞ M M (λn ) = N (t). 0 (b) If N ∈ CM,λ then, for every fixed t > 0, there exists a sequence of functions {Mn } ⊂ CM,λ such that N (t) = limn→∞ Mn (t).

The following classical result – in the form we present herein – is a particular case of [10, Proposition 2]: Proposition 2.10 (K. Lindberg). Let hM be an Orlicz sequence space, and let (un )∞ n=1 be a normalized block ∞ basic sequence of the unit vector basis of hM . Then there exists a subsequence (unk )∞ k=1 of (un )n=1 , which is equivalent to the unit vector basis of hN , for some (possibly degenerate) Orlicz function N ∈ CM,1 . Moreover, 0 the function N is the pointwise limit of a sequence {Mk } ⊂ CM,1 , where Mk ∈ CM,u , for every k ≥ 1. n ∞ k

The following proposition describes the set of symmetric vanishing block bases with respect to the unit vector basis of M , in the case the function M fulfills the Δ2 -condition at zero (cf., [10, Proposition 2 and Lemma 4]).

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Proposition 2.11. Let M be an Orlicz function fulfilling the Δ2 -condition at zero and let M be its associated Orlicz sequence space. Let (un )∞ n=1 be a vanishing block basis of the unit vector basis for M . Then, there exists a subsequence of (unk )∞ k=1 which is equivalent to the unit vector basis of N , for some Orlicz function (0) N ∈ CM . Moreover, the function N is the pointwise limit of a sequence {Mk } ⊂ CM,1 , where Mk ∈ 0 CM,un ∞ , for every k ≥ 1. k

A ϕ-function M (i.e., M is increasing and continuous on [0, ∞) with M (0) = 0 and lims→∞ M (s) = ∞), (ts) is regularly varying at zero (cf., [14, p. 11], [4, Section 2.3., p. 83]), if the limit f (s) = limt→0+ M M (t) exists and is non-zero, for each s > 0. For example, Mp (t) := tp is regularly varying for every p > 0. If the limit (ts) f (s) = limt→0+ M M (t) exists and f (t) = M∞ (t) := 0, if 0 ≤ t < 1, 1 if t = 1, and ∞ if t > 1, then the function M is rapidly varying (or rapidly increasing) at zero. Notice that limp→∞ Mp (t) = M∞ (t), for every t ≥ 0. The notions of regularly varying and rapidly increasing Orlicz functions fit into our approach due to the following fact (cf., [8, Lemma 4.4]): (st) Proposition 2.12. Let M be an Orlicz function such that the limit lims→0+ M M (s) = f (t) exists. Then either (a) f (t) > 0, for some 0 < s < 1, in which case M is a regularly varying function and f (t) = tp , t ≥ 0,  (s) where p = lims→0+ sM M (s) , or (b) f (s) = 0, for some 0 < s < 1, in which case M is rapidly varying, that is, f (t) = M∞ (t), t ≥ 0, where M∞ (t) denotes the pointwise limit of Mp (t) = tp as p → ∞.

The sets CM , where M is either regularly varying or rapidly varying, have been computed in [8, Lemma 4.3]. Proposition 2.13. Let M be a non-degenerate Orlicz function such that the limit f (t) = lims→0+ for every t ≥ 0. Then CM :=



  CM,λ = Mp (t) ,

where p = lim+ s→0

0<λ≤1

M (st) M (s)

exists,

log M (s) . log s

3. Concrete examples Let a, b, p be fixed real numbers, with a = 0, b > 0, and p > 1. Let Ma,b,p be the function defined on the interval [0, 1/b) by the formulae Ma,b,p (0) = 0, and a  Ma,b,p (x) = xp log(bx)

for x = 0.

This family has been studied in [9, Example 3.2]. It is proved there that, for some 0 < t0 < 1, Ma,b,p is a strictly increasing, continuous convex function on [0, t0 ] and extends – linearly on [t0 , ∞) – to an Orlicz function, denoted also by Ma,b,p . Also, for x ∈ (0, t0 ), we have that xMa,b,p (x) a =p− , Ma,b,p (x) | log(bx)| xM 

(x)

a,b,p whence limx→0 Ma,b,p (x) = p. For brevity, we put M = Ma,b,p , and consider the function f defined for every t ∈ (−∞, log t0 ] by f (t) = log M (et ). Hence,



 a

f (t) = log M et = log ept log bet  = pt + a log | log b + t|,

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and so

f (t) = pt + a log | log b + t| = p +

a , log b + t

and f (t) = −a/(log b + t)2 . Thus, f is concave for a > 0 and convex for a < 0. Let us define the Orlicz function N by the formula N (x) := M (t0 x)/M (t0 ), x ≥ 0. Then, N is geometrically concave for a > 0 and geometrically convex for a < 0. 4. The proofs Proof of Theorem 1.2. Passing to an equivalent norm, if necessary, we assume that the symmetric basis constant of (xn ) equals 1. Let (yn ) be a symmetric basic sequence in X = [xn ]. If (yn ) is equivalent to the unit vector basis of 1 then it dominates every normalized monotone basis and in particular all the normalized block basis of (xn ). In this case the statement is trivially true. Assume otherwise that (yn ) is not equivalent to the unit vector basis of 1 . Then, by Proposition 2.4, the sequence (yn ) is weakly null, and according to Proposition 2.3, we conclude that some subsequence of (yn ) is equivalent to a normalized block basis of (xn ). By passing to an equivalent basis, if necessary, we may assume that (yn ) itself is a normalized block basis of (xn ), say, 

pn+1

yn =

αk xk ,

k=pn +1

with yn  = 1 for every n ≥ 1. Also, after rearranging the symmetric basis (xn ), we assume that, for each n ≥ 1, the (finite) sequence consisting of the non-zero coefficients of yn is decreasing. Now, there are two mutually exclusive cases: Case 1 M := supn≥1 {pn+1 − pn } < ∞, Case 2 supn≥1 {pn+1 − pn } = ∞. In Case 1, we use Proposition 2.6 in order to conclude that (yn ) is equivalent to (xn ), which is in turn equivalent to any basic sequence generated by the vector α = x1 . In Case 2, after switching to a subsequence of (yn ), if necessary, and then, again if necessary, after rearranging the terms of the supports of the remaining blocks in such a way that they are consecutive, we may assume that (pn+1 − pn ) is an increasing unbounded sequence. Now there are two possibilities: (a) either for every ε > 0, there exists an integer N = Nε such that, for every n for which pn+1 − pn ≥ N , pn+1 we have  k=p αk xk  < ε, or n +N (b) there exists an ε0 > 0, such that, for every positive integer N , there exists n = n(N ) such that pn+1 (pn+1 − pn ) ≥ N and  k=p αk xk  ≥ ε0 . n +N In case (a), for ε > 0 fixed and every l ≥ 1, we apply the condition for ε/2l+1 in order to obtain an increasing sequence {Nl }∞ l=1 of natural numbers such that, for every n for which pn+1 − pn ≥ Nl , we have     



pn+1

k=pn +Nl

  ε  αk xk  < l+1 .  2

(3)

Since pn+1 − pn is increasing on n, for every l ≥ 1 there exists an integer nl such that pn+1 − pn ≥ Nl , for every n ≥ nl . On the other hand, since the basis (xn ) is symmetric, for every l ≥ 1 we obtain

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   yn − 



pn+1

k=pn +Nl

  p n+1      αk xk  =  αk xk −   k=pn +1



pn+1

k=pn +Nl

387

   αk xk  

p +N −1  N −1  l  n l    ε     = αk xk  =  αk+pn xk  < l+1 .     2 k=pn +1

k=1

Since yn  = 1, from inequality (3) applied to the latter inequality, we obtain 1−

ε 2l+1

N −1  l   ε   ≤ αk+pn xk  ≤ 1 + l+1 ,   2

(4)

k=1

for every l ≥ 1 and n ≥ nl (for which pn+1 − pn ≥ Nl ). In particular, for l = 1, the sequence N 1 −1

αk+pn xk ,

n ≥ n1 ,

k=1

is norm-bounded in the N1 -dimensional subspace of X. Consequently, this sequence has a convergent subsequence to, say, N 1 −1

A k xk ,

k=1 ∞ where A1 , A2 , . . . , AN1 are fixed scalars. Thus, there is a subsequence {nj,1 }∞ j=1 of {n}n=n1 such that, for every 1 ≤ k ≤ N1 − 1,

lim αk+pnj,1 = Ak .

j−→∞

Hence, since the basis (xn ) is symmetric (and we may assume it is 1-symmetric), for every m ∈ {nj,1 }, we obtain  p +N −1  N −1 pm +N N 1 1 −1   m 1   1 −1     αk+pm xk − A k xk  =  αk xk − Ak−pm xk  < ε/4.      k=1

k=1

k=pm +1

k=pm +1

Now we apply the same procedure for N2 > N1 and the sequence subsequence {nj,2 } of {nj,1 }, such that, for every m ∈ {nj,2 },

N2 −1 k=1

αk+pnj,1 xk , and we obtain a

 p +N −1 pm +N   m 2 2 −1   αk xk − Ak−pm xk  < ε/8,    k=pm +1

k=pm +1

where limm−→∞ αk+pm = Ak , for every 1 ≤ k ≤ N2 −1. Let us observe that the numbers Ak , for 1 ≤ k ≤ N1 , are the same as before, because the limit of the k-th coefficient must coincide in every subsequence. Continuing this way, we inductively pick subsequences {nj,l } such that, for every l ≥ 1, {nj,l } ⊂ {nj+1,l }, pm+1 − pm ≥ Nl , and  p +N −1 pm +N   m l l −1   αk xk − Ak−pm xk  < ε/2l+1 ,    k=pm +1

k=pm +1

(5)

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388

for every m ∈ {nj,l }, and where limm−→∞ αk+pm = Ak , for every 1 ≤ k ≤ Nl −1. Now consider the sequence ml = nl,l , l = 1, 2, . . . . Set pml +Nl −1

zl :=



pml +Nl −1

αk xk ,



and wl :=

k=pml +1

Ak−pml xk ,

k=pml +1

for every l ≥ 1, and also notice that yml =

pml +1 k=pml +1

αk xk , for every l ≥ 1.

Let us observe that the inequality (3) implies that yml − zl  < ε/2l+1 , while by using the inequality (5) we have that wl − zl  < ε/2l+1 and, consequently yml − wl  < ε/2l . Since (yn ) is a 1-unconditional basis (because (xn ) is 1-symmetric), we have that the sequence of functionals biorthogonal to (yml ) is normalized and, consequently, ∞ 

∗ yml − wl ym ≤ l

l=1

∞  ε = ε. 2l l=1

Therefore, Proposition 2.2 applies to conclude that the sequence (yml ) is On the other hand, by inequalities (4) and (5), we obtain





1 − ε/2l+1 ≤ zl  ≤ 1 + ε/2l+1

1+ε 1−ε -equivalent

to (wl ).

and wl − zl  < ε/2l+1 ,

which implies that





1 − ε/2l ≤ wl  ≤ 1 + ε/2l .

By using again the symmetry of the basis (xn ), we obtain that, for every l = 1, 2, . . . , N −1  l 



  l Ak xk  ≤ 1 + ε/2l , 1 − ε/2 ≤   

(6)

k=1

Nl −1 for every l ≥ 1. In particular, the sequence k=1 Ak xk , l = 1, 2, . . . , is norm-bounded. Since the basis (xn ) ∞ is boundedly complete, the series k=1 Ak xk converges to a vector α in X. By inequalities (6), we have



1 − ε/2l ≤ α ≤ 1 + ε/2l ,

for every l ≥ 1, and hence α is a norm-one vector of X. On the other hand, by Proposition 2.5, for every δ > 0, there is a subsequence of (wl ) which is (1 + ∞ δ)-equivalent to a basic sequence generated by the norm-one vector α = k=1 Ak xk . Since the subsequence 1+ε 1+ε (yml ) is 1−ε -equivalent to (wl ), there exists a subsequence of (yml ), and hence of (yn ), which is 1−ε (1 + ∞ δ)-equivalent to a basic sequence generated by the norm-one vector α = k=1 Ak xk . This proves the statement of Theorem 1.2 for case (a). In case (b), the vector 

pn+1

vn =

k=pn +N

αk xk ,

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satisfies the inequality ε ≤ vn  ≤ 1, so the block basis (vn ) of (xn ) is semi-normalized. We claim that (yn ) dominates (vn ). Indeed, on one hand, since the coefficients of (yn ) are decreasing, we have that maxpn +N ≤k≤pn+1 αk < αpn +N −1 . On the other hand, p +N −1  p +N −1   n   n      1≥ αk xk  ≥  xk αpn +N −1 ,     k=1

k=1

pn +N −1 xk  is bounded, in which case, by [16, Lemma 2.3], (xk ) is equivalent which implies that either  k=1 to the unit vector basis of c0 , or αpn +N −1 , and hence maxpn +N ≤k≤pn+1 αk , tends to zero as N → ∞. But c0 does not embed in X because (xn ) is boundedly complete (cf., [12, Theorem 1.c.10, p. 22]). This forces the second possibility: the sequence of non-zero coefficients of (vn ) tends to zero; that is, (vn ) is a (semi-normalized) block basis of (xn ) having coefficients tending to zero. Since (xn ) is unconditional, and (vn ) is obtained by deleting coefficients from (yn ), we conclude that (yn ) dominates (vn ). 2 In order to proof Theorem 1.3, we will use the following lemma. It gathers the main properties of geometrically convex Orlicz functions that we will need. Lemma 4.1. Let M be an Orlicz function which is geometrically convex on [0, δ], for some δ > 0. Then: (i) The function M fulfills the Δ2 -condition at zero and, consequently, the unit vectors form a boundedly complete symmetric basis for the Orlicz space M . (st) p (ii) The function M is regularly varying in the sense that lims→0+ M M (s) = t , for every t ≥ 0, where M (s) p = lims→0+ loglog s . (iii) For the same p as in part (ii), every normalized basic sequence in M has a subsequence which is dominated by the unit vector basis of p . In particular, every (sub)symmetric basic sequence in M is dominated by the unit vector basis of p .

Proof. (i) Let δ > 0 be such that M is geometrically convex on [0, δ]. By Proposition 2.1, the function (δt) M (δst) M (δs) M (δt) t → M M (δ) is super-multiplicative on [0, 1], that is, M (δ) ≥ M (δ) M (δ) , for every s, t ∈ [0, 1], or, equivalently, M (δ)M (δst) ≥ M (δs)M (δt), for every s, t ∈ [0, 1]. By putting 2x = δt and s = 12 , we obtain that M (δ) M (x) ≥ M (2x), for every x ∈ [0, δ/2]; that is, M fulfills the Δ2 -condition on [0, δ/2]. Finally, by M ( δ2 ) Proposition 2.7, the unit vectors form a boundedly complete symmetric basis for the space M . (xyz) M (yz) (ii) By Proposition 2.1 again, we have the inequality M M (xz) ≥ M (z) , for every x, y ∈ (0, 1), z ∈ (0, δ), which in turn implies that M (ts ) M (ts) ≥ , M (s) M (s )

whenever 0 < s < s < δ, t ∈ (0, 1).

It follows that, for every fixed t ∈ (0, 1), the function s → lims→0+

M (st) M (s)

M (st) M (s)

(7)

is decreasing on (0, δ), and hence 0 <

(st) = sup0
M is regularly varying, that is,

M (s) the limit regularity p = lims→0+ loglog s . (iii) According to part (i), M fulfills the Δ2 -condition at 0 and, consequently, the unit vector forms a boundedly complete symmetric basis for M = hM . According to Proposition 2.10, every normalized basic sequence in M has a subsequence which is equivalent to the unit vector basis of hN , for some N ∈ CM,1 . Therefore, by Proposition 2.8, it is enough to prove that tp dominates every function in CM,1 . By inequality (7), for every fixed t ∈ (0, 1), the function s → M (ts)/M (s) is decreasing on (0, δ). Also, by part (ii), M is regularly varying, and hence

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Ms (t) :=

M (ts) M (s t) ≤  lim + = tp , s −→0 M (s ) M (s)

(8)

for every 0 < t ≤ 1. On the other hand, according to Proposition 2.9, for every N ∈ EM,1 and for every fixed t ∈ (0, 1), there (λn t) exists a sequence {λn } ⊂ (0, 1) such that and limn→∞ M M (λn ) = N (t). Thus, by using (8), we obtain that M (λn t) ≤ tp . n→∞ M (λn )

N (t) = lim

Summing up, we have proved that N (t) ≤ tp , for every N ∈ EM,1 . On the other hand, every function in CM,1 is a pointwise limit of convex combinations of the functions in EM,1 and, consequently, tp ≥ N (t), for every N ∈ CM,1 . Now, according to the equivalence of (ii) and (iii) of Proposition 2.8, the unit vector basis of M dominates the unit vector basis of hN for every N ∈ CM,1 . This completes the proof of Lemma 4.1. 2 Proof of Theorem 1.3. According to Lemma 4.1(i), M fulfills the Δ2 -condition at 0 and, consequently, the unit vectors form a boundedly complete symmetric basis for M = hM , denoted (eM n ). Let (xn ) be a symmetric basic sequence in M . By Theorem 1.2, (xn ) is • equivalent to a basic sequence generated by a vector with respect to the basis (eM n ), M • or dominates a vanishing block basis (yn ), of (en ). Therefore, we have to consider only the latter case. By Lemma 4.1(iii), (xn ) is dominated by the unit vector M (s) basis (epn ) of p , where p is the index of regularity for M , that is, p = lims→0+ loglog s . Claim. The sequence (yn ) has a subsequence (ynk ) equivalent to the basis (epn ), where again p = M (s) lims→0+ loglog s . By Lemma 4.1(ii), M is regularly varying and, by applying Proposition 2.13, we conclude that CM = {tp }, where p is the index of regularity for M . By Proposition 2.11, there is a subsequence (ynk ) of (yn ) equivalent to the unit vector basis of hN , for some N ∈ CM . Hence, N (t) = tp , and therefore the sequence (ynk ) is equivalent to the unit vector basis of p , as claimed. Since the basic sequence (xn ) is symmetric, from the Claim we obtain that (xn ) dominates the unit vector basis of p . Summing up, we have proved that (epn ) dominates (xn ) and (xn ) dominates (epn ), where p is the index of regularity for M , which implies that (xn ) and (epn ) are equivalent. This concludes the proof. 2 Acknowledgments The material presented here, is a part of the second named author’s Doctoral Thesis under the supervision of Professor Marek Wójtowicz (Advisor) and, first named author (Co-Advisor). Both authors are deeply indebted to Prof. Wójtowicz for his valuable remarks and his support during the course of this research. We also wish to thank the referee, for his comments and suggestions which improved the presentation of the paper considerably. References [1] Z. Altshuler, P.G. Casazza, B.L. Lin, On symmetric basic sequences in Lorentz sequence spaces, Israel J. Math. 15 (1973) 140–155.

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