AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation86537110.1155/2009/865371865371Research ArticleOn Perfectly Homogeneous Bases in Quasi-Banach SpacesAlbiacF.LeránozC.ReichSimeonDepartamento de MatemáticasUniversidad Pública de Navarra31006 PamplonaSpainunavarra.es200916062009200922042009030620092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

For 0<p< the unit vector basis of p has the property of perfect homogeneity: it is equivalent to all its normalized block basic sequences, that is, perfectly homogeneous bases are a special case of symmetric bases. For Banach spaces, a classical result of Zippin (1966) proved that perfectly homogeneous bases are equivalent to either the canonical c0-basis or the canonical p-basis for some 1p<. In this note, we show that (a relaxed form of) perfect homogeneity characterizes the unit vector bases of p for 0<p<1 as well amongst bases in nonlocally convex quasi-Banach spaces.

1. Introduction and Background

Let us first review the relevant elementary concepts and definitions. Further details can be found in the books [1, 2] and the paper . A (real) quasi-normed space X is a locally bounded topological vector space. This is equivalent to saying that the topology on X is induced by a quasi-norm , that is, a map ·:X[0,) satisfying

x=0 if and only if x=0;

αx=|α|x if α,xX;

there is a constant κ1 so that for any x1 and x2X we have x1+x2κ(x1+x2).

The best constant κ in inequality (1.1) is called the modulus of concavity of the quasi-norm. If κ=1, the quasi-norm is a norm. A quasi-norm on X is p-subadditive if x1+x2px1p+x2p,x1,x2X. A theorem by Aoki  and Rolewicz  asserts that every quasi-norm has an equivalent p-subadditive quasi-norm, where 0<p1 is given by κ=21/p-1. A p-subadditive quasi-norm · induces an invariant metric on X by the formula d(x,y)=x-yp. The space X is called quasi-Banach space if X is complete for this metric. A quasi-Banach space is isomorphic to a Banach space if and only if it is locally convex.

A basis (xn)n=1 of a quasi-Banach space X is symmetric if (xn)n=1 is equivalent to (xπ(n))n=1 for any permutation π of . Symmetric bases are unconditional and so there exists a nonnegative constant K such that for all x=n=1anxn the inequality n=1θnanxnKn=1anxn holds for any bounded sequence (θn)n=1B. The least such constant K is called the unconditional constant of (xn)n=1.

For instance, the canonical basis of the spaces p for 0<p< is symmetric and 1-unconditional. What is more, it is the only symmetric basis of p up to equivalence, that is, whenever (xn)n=1 is another normalized symmetric basis of p, there is a constant C such that 1C(n=1|an|p)1/pn=1anxnC(n=1|an|p)1/p, for any finitely nonzero sequence of scalars (an)n=1 [6, 7].

The spaces p for 0<p<1 share the property of uniqueness of symmetric basis with all natural quasi-Banach spaces whose Banach envelope (i.e., the smallest containing Banach space) is isomorphic to 1, as was recently proved in . For other results on uniqueness of unconditional or symmetric basis in nonlocally convex quasi-Banach spaces the reader can consult the papers [9, 10].

This article illustrates how Zippin’s techniques can also be used to characterize the unit vector bases of p for 0<p<1 as the only, up to equivalence, perfectly homogeneous bases in nonlocally convex quasi-Banach spaces. We use standard Banach space theory terminology and notation throughout, as may be found in [11, 12].

2. Perfectly Homogeneous Bases in Quasi-Banach Spaces

Let (xi)i=1 be a basis for a quasi-Banach space X. A block basic sequence (un)n=1 of (xi)i=1, un=pn-1+1pnaixi, is said to be a constant coefficient block basic sequence if for each n there is a constant cn so that ai=cn or ai=0 for pn-1+1ipn.

Definition 2.1.

A basis (xi)i=1 of a quasi-Banach space X is almost perfectly homogeneous if every normalized constant coefficient block basic sequence of (xi)i=1 is equivalent to (xi)i=1.

Let us notice that using a uniform boundedness argument we obtain that, in fact, if (xi)i=1 is almost perfectly homogeneous then it is uniformly equivalent to all its normalized constant coefficient block basic sequences. That is, there is a constant M1 such that for any normalized constant coefficient block basic sequence (un)n=1 of (xi)i=1 we have M-1k=1nakxkk=1nakukMk=1nakxk, for all choices of scalars (ak)k=1n and n. Equation (2.2) also yields that for any increasing sequence of integers (kj)j=1, M-1j=1nxjj=1nxkjMj=1nxj.

This is our main result (cf. ).

Theorem 2.2.

Let X be a nonlocally convex quasi-Banach space with normalized basis (xi)i=1. Suppose that (xi)i=1 is almost perfectly homogeneous. Then (xi)i=1 is equivalent to the canonical basis of q for some 0<q<1.

Proof.

Let κ be the modulus of concavity of the quasi-norm. Since X is nonlocally convex, κ>1. By the Aoki-Rolewicz theorem we can assume that the quasi-norm is p-subadditive for 0<p<1 such that κ=21/p-1. We will show that (xi)i=1 is equivalent to the canonical q-basis for some pq<1.

By renorming, without loss of generality we can assume (xi)i=1 to be 1-unconditional. For each n put, λ(n)=i=1nxi. Note that 1λ(n)n1/p,n, and that, by the 1-unconditionality of the basis, the sequence (λ(n))n=1 is nondecreasing.

We are going to construct disjoint blocks of length n of the basis (xi)i=1 as follows: v1=i=1nxi,v2=i=n+12nxi,,vj=i=(j-1)n+1jnxi,. Equation (2.3) says that M-1λ(n)vjMλ(n),j, and so by the 1-unconditionality of (xi)i=1, 1Mλ(n)j=1mvjj=1mvj-1vjMλ(n)j=1mvj,m. On the other hand, by (2.2) we know that λ(m)Mj=1mvj-1vjMλ(m),m. If we put these last two inequalities together we obtain 1M2λ(m)λ(n)λ(mn)M2λ(m)λ(n),m,n. Substituting in (2.10) integers of the form m=2k and n=2j give 1M2λ(2k)λ(2j)λ(2j+k)M2λ(2k)λ(2j),k,j. For k=0,1,2,, let h(k)=log2λ(2k). From (2.11) it follows that |h(j)-h(k)-h(j+k)|2log2M. We need the following well-known lemma from real analysis.

Lemma 2.3.

Suppose that (sn)n=1 is a sequence of real numbers such that |sm+n-sm-sn|1 for all m,n. Then there is a constant c so that |sn-cn|1,n=1,2,.

Lemma 2.3 yields a constant c so that |h(k)-ck|2log2M,k=1,2,. In turn, using (2.5) we have 1λ(2k)2k/p,k=1,2, which implies 0h(k)kp, and so, combining with (2.15) we obtain that the range of possible values for c is 0c1p. If c=0 then (λ(n))n=1 would be (uniformly) bounded and so (xi)i=1 would be equivalent to the canonical basis of c0, a contradiction with the local nonconvexity of X. Otherwise, if 0<c1/p there is q[p,) such that c=1/q. This way we can rewrite (2.15) in the form |h(k)-kq|2log2M,k, or equivalently, M-22k/qλ(2k)2k/qM2,k. Now, given n we pick the only integer k so that 2k-1n2k. Then, λ(2k-1)λ(n)λ(2k), and so M-22-1/qn1/qλ(n)M221/qn1/q. If A is any finite subset of , by (2.3) we have M-1λ(|A|)jAxjMλ(|A|), hence C-1|A|1/qjAxjC|A|1/q, where C=M321/q.

To prove the equivalence of (xi)i=1 with the canonical basis of q, given any n we let (ai)i=1n be nonnegative scalars such that aiq and i=1naiq=1. Each aiq can be written in the form aiq=mi/m where mi, m is de common denominator of the aiq's, and i=1nmi=m.

Let A1 be interval of natural numbers [1,m1] and for j=2,,n let Ai be the interval of natural numbers [m1++mi-1+1,m1++mi]. The sets A1,,An are disjoint and have cardinality |Ai|=mi for each i=1,,n. Consider the normalized constant coefficient block basic sequence defined as ui=ci-1jAixj,1in, where ci=jAixk. Equation (2.24) yields C-1mi1/qciCmi1/q,1in. Therefore, C-1m1/qi=1njAixji=1naiuiCm1/qi=1njAixk, that is, C-1λ(m)m1/qi=1naiuiCλ(m)m1/q. Thus, 1C2Mi=1naiuiC2M. Using (2.2) again, we have 1C2M2i=1naixiC2M2. We note that a simple density argument shows that (2.30) holds whenever i=1n|ai|q=1 (i.e., without the assumption that |ai|q is rational), and this completes the proof that (xi)i=1 is equivalent to the canonical q-basis for some pq<. Since X is not locally convex, we conclude that pq<1.

Acknowledgment

The authors would like to acknowledge support from the Spanish Ministerio de Educación y Ciencia Research Project Espacios Topológicos Ordenados: Resultados Analíticos y Aplicaciones Multidisciplinares, reference number MTM2007-62499.

KaltonN. J.PeckN. T.RogersJ. W.An F-Space Sampler198589Cambridge, UKCambridge University PressLondon Mathematical Society Lecture NoteRolewiczS.Metric Linear Spaces1985202ndDordrecht, The NetherlandsD. Reidelxii+459Mathematics and Its Applications (East European Series)MR808176KaltonN. J.Quasi-Banach spacesHandbook of the Geometry of Banach Spaces, Vol. 22003Amsterdam, The NetherlandsNorth-Holland10991130MR199919210.1016/S1874-5849(03)80032-3ZBL1059.46004AokiT.Locally bounded linear topological spacesProceedings of the Imperial Academy, Tokyo194218588594MR0014182ZBL0060.26503RolewiczS.On a certain class of linear metric spacesBulletin de L'Académie Polonaise des Sciences19575471473MR0088682ZBL0079.12602KaltonN. J.Orlicz sequence spaces without local convexityMathematical Proceedings of the Cambridge Philosophical Society1977812253277MR043319410.1017/S0305004100053342ZBL0345.46013LindenstraussJ.TzafririL.On Orlicz sequence spacesIsrael Journal of Mathematics197110379390MR031378010.1007/BF02771656ZBL0227.46042AlbiacF.LeránozC.Uniqueness of symmetric basis in quasi-Banach spacesJournal of Mathematical Analysis and Applications200834815154MR244932610.1016/j.jmaa.2008.07.011AlbiacF.LeránozC.Uniqueness of unconditional basis in Lorentz sequence spacesProceedings of the American Mathematical Society2008136516431647MR237359310.1090/S0002-9939-08-09222-8ZBL1140.46002KaltonN. J.LeránozC.WojtaszczykP.Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spacesIsrael Journal of Mathematics1990723299311MR112022310.1007/BF02773786ZBL0753.46013AlbiacFKaltonN. J.Topics in Banach Space Theory2006233New York, NY, USASpringerxii+373Graduate Texts in MathematicsMR2192298ZBL1087.65048LindenstraussJ.TzafririL.Classical Banach Spaces. I: Sequence Spaces1977Berlin, GermanySpringerxiii+188Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 9MR0500056ZippinM.On perfectly homogeneous bases in Banach spacesIsrael Journal of Mathematics19664265272MR020980910.1007/BF02771642ZBL0148.11202