On Perfectly Homogeneous Bases in Quasi-Banach Spaces

Abstract

For 0<𝑝<∞ the unit vector basis of ℓ_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 ℓ_𝑝-basis for some 1≤𝑝<∞. In this note, we show that (a relaxed form of) perfect homogeneity characterizes the unit vector bases of ℓ_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 [3]. 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

(i)‖𝑥‖=0 if and only if 𝑥=0;(ii)‖𝛼𝑥‖=|𝛼|‖𝑥‖ if 𝛼∈ℝ,𝑥∈𝑋;(iii) there is a constant 𝜅≥1 so that for any 𝑥₁ and 𝑥₂∈𝑋 we have ‖𝑥₁+𝑥₂‖≤𝜅(‖𝑥₁‖+‖𝑥₂‖). (1.1)

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+𝑥₂‖^𝑝≤‖𝑥₁‖^𝑝+‖𝑥₂‖^𝑝,𝑥₁,𝑥₂∈𝑋. (1.2) A theorem by Aoki [4] and Rolewicz [5] asserts that every quasi-norm has an equivalent 𝑝-subadditive quasi-norm, where 0<𝑝≤1 is given by 𝜅=2^1/𝑝−1. A 𝑝-subadditive quasi-norm ‖⋅‖ induces an invariant metric on 𝑋 by the formula 𝑑(𝑥,𝑦)=‖𝑥−𝑦‖^𝑝. 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 (𝑥_𝑛)^∞_𝑛=1 of a quasi-Banach space 𝑋 is symmetric if (𝑥_𝑛)^∞_𝑛=1 is equivalent to (𝑥_𝜋(𝑛))^∞_𝑛=1 for any permutation 𝜋 of ℕ. Symmetric bases are unconditional and so there exists a nonnegative constant 𝐾 such that for all 𝑥=∑^∞_𝑛=1𝑎_𝑛𝑥_𝑛 the inequality ‖^∞_𝑛=1𝜃_𝑛𝑎_𝑛𝑥_𝑛‖≤𝐾‖^∞_𝑛=1𝑎_𝑛𝑥_𝑛‖ (1.3) holds for any bounded sequence (𝜃_𝑛)^∞_𝑛=1∈𝐵_ℓ∞. The least such constant 𝐾 is called the unconditional constant of (𝑥_𝑛)^∞_𝑛=1.

For instance, the canonical basis of the spaces ℓ_𝑝 for 0<𝑝<∞ is symmetric and 1-unconditional. What is more, it is the only symmetric basis of ℓ_𝑝 up to equivalence, that is, whenever (𝑥_𝑛)^∞_𝑛=1 is another normalized symmetric basis of ℓ_𝑝, there is a constant 𝐶 such that 1𝐶(^∞_𝑛=1|𝑎_𝑛|^𝑝)^1/𝑝≤‖^∞_𝑛=1𝑎_𝑛𝑥_𝑛‖≤𝐶(^∞_𝑛=1|𝑎_𝑛|^𝑝)^1/𝑝, (1.4) for any finitely nonzero sequence of scalars (𝑎_𝑛)^∞_𝑛=1 [6, 7].

The spaces ℓ_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 ℓ₁, as was recently proved in [8]. 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 ℓ_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 (𝑥_𝑖)^∞_𝑖=1 be a basis for a quasi-Banach space 𝑋. A block basic sequence (𝑢_𝑛)^∞_𝑛=1 of (𝑥_𝑖)^∞_𝑖=1, 𝑢_𝑛=^𝑝𝑛_{𝑝𝑛−1+1}𝑎_𝑖𝑥_𝑖, (2.1) is said to be a constant coefficient block basic sequence if for each 𝑛 there is a constant 𝑐_𝑛 so that 𝑎_𝑖=𝑐_𝑛 or 𝑎_𝑖=0 for 𝑝_𝑛−1+1≤𝑖≤𝑝_𝑛.

Definition 2.1. A basis (𝑥_𝑖)^∞_𝑖=1 of a quasi-Banach space 𝑋 is almost perfectly homogeneous if every normalized constant coefficient block basic sequence of (𝑥_𝑖)^∞_𝑖=1 is equivalent to (𝑥_𝑖)^∞_𝑖=1.

Let us notice that using a uniform boundedness argument we obtain that, in fact, if (𝑥_𝑖)^∞_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 𝑀≥1 such that for any normalized constant coefficient block basic sequence (𝑢_𝑛)^∞_𝑛=1 of (𝑥_𝑖)^∞_𝑖=1 we have 𝑀⁻¹‖^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘‖≤‖^𝑛_𝑘=1𝑎_𝑘𝑢_𝑘‖≤𝑀‖^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘‖, (2.2) for all choices of scalars (𝑎_𝑘)^𝑛_𝑘=1 and 𝑛∈ℕ. Equation (2.2) also yields that for any increasing sequence of integers (𝑘_𝑗)^∞_𝑗=1, 𝑀⁻¹‖^𝑛_𝑗=1𝑥_𝑗‖≤‖^𝑛_𝑗=1𝑥_𝑘𝑗‖≤𝑀‖^𝑛_𝑗=1𝑥_𝑗‖. (2.3)

This is our main result (cf. [13]).

Theorem 2.2. Let 𝑋 be a nonlocally convex quasi-Banach space with normalized basis (𝑥_𝑖)^∞_𝑖=1. Suppose that (𝑥_𝑖)^∞_𝑖=1 is almost perfectly homogeneous. Then (𝑥_𝑖)^∞_𝑖=1 is equivalent to the canonical basis of ℓ_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=2^1/𝑝−1. We will show that (𝑥_𝑖)^∞_𝑖=1 is equivalent to the canonical ℓ_𝑞-basis for some 𝑝≤𝑞<1.
By renorming, without loss of generality we can assume (𝑥_𝑖)^∞_𝑖=1 to be 1-unconditional. For each 𝑛 put, 𝜆(𝑛)=‖^𝑛_𝑖=1𝑥_𝑖‖. (2.4) Note that 1≤𝜆(𝑛)≤𝑛^1/𝑝,𝑛∈ℕ, (2.5) 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 (𝑥_𝑖)^∞_𝑖=1 as follows: 𝑣₁=^𝑛_𝑖=1𝑥_𝑖,𝑣₂=^2𝑛_𝑖=𝑛+1𝑥_𝑖,…,𝑣_𝑗=^𝑗𝑛_{𝑖=(𝑗−1)𝑛+1}𝑥_𝑖,…. (2.6) Equation (2.3) says that 𝑀⁻¹𝜆(𝑛)≤‖𝑣_𝑗‖≤𝑀𝜆(𝑛),𝑗∈ℕ, (2.7) and so by the 1-unconditionality of (𝑥_𝑖)^∞_𝑖=1, 1‖𝑀𝜆(𝑛)^𝑚_𝑗=1𝑣_𝑗‖≤‖^𝑚_𝑗=1‖𝑣_𝑗‖⁻¹𝑣_𝑗𝑀‖≤‖𝜆(𝑛)^𝑚_𝑗=1𝑣_𝑗‖,𝑚∈ℕ. (2.8) On the other hand, by (2.2) we know that 𝜆(𝑚)𝑀≤‖^𝑚_𝑗=1‖𝑣_𝑗‖⁻¹𝑣_𝑗‖≤𝑀𝜆(𝑚),𝑚∈ℕ. (2.9) If we put these last two inequalities together we obtain 1𝑀²𝜆(𝑚)𝜆(𝑛)≤𝜆(𝑚𝑛)≤𝑀²𝜆(𝑚)𝜆(𝑛),𝑚,𝑛∈ℕ. (2.10) Substituting in (2.10) integers of the form 𝑚=2^𝑘 and 𝑛=2^𝑗 give 1𝑀²𝜆(2^𝑘)𝜆(2^𝑗)≤𝜆(2^𝑗+𝑘)≤𝑀²𝜆(2^𝑘)𝜆(2^𝑗),𝑘,𝑗∈ℕ. (2.11) For 𝑘=0,1,2,…, let ℎ(𝑘)=log₂𝜆(2^𝑘). From (2.11) it follows that |ℎ(𝑗)−ℎ(𝑘)−ℎ(𝑗+𝑘)|≤2log₂𝑀. (2.12) We need the following well-known lemma from real analysis.
Lemma 2.3. Suppose that (𝑠_𝑛)^∞_𝑛=1 is a sequence of real numbers such that |𝑠_𝑚+𝑛−𝑠_𝑚−𝑠_𝑛|≤1 (2.13) for all 𝑚,𝑛∈ℕ. Then there is a constant 𝑐 so that |𝑠_𝑛−𝑐𝑛|≤1,𝑛=1,2,…. (2.14)
Lemma 2.3 yields a constant 𝑐 so that |ℎ(𝑘)−𝑐𝑘|≤2log₂𝑀,𝑘=1,2,…. (2.15) In turn, using (2.5) we have 1≤𝜆(2^𝑘)≤2^𝑘/𝑝,𝑘=1,2,… (2.16) which implies 𝑘0≤ℎ(𝑘)≤𝑝, (2.17) and so, combining with (2.15) we obtain that the range of possible values for 𝑐 is 10≤𝑐≤𝑝. (2.18) If 𝑐=0 then (𝜆(𝑛))^∞_𝑛=1 would be (uniformly) bounded and so (𝑥_𝑖)^∞_𝑖=1 would be equivalent to the canonical basis of 𝑐₀, a contradiction with the local nonconvexity of 𝑋. Otherwise, if 0<𝑐≤1/𝑝 there is 𝑞∈[𝑝,∞) such that 𝑐=1/𝑞. This way we can rewrite (2.15) in the form 𝑘|ℎ(𝑘)−𝑞|≤2log₂𝑀,𝑘∈ℕ, (2.19) or equivalently, 𝑀⁻²2^𝑘/𝑞≤𝜆(2^𝑘)≤2^𝑘/𝑞𝑀²,𝑘∈ℕ. (2.20) Now, given 𝑛∈ℕ we pick the only integer 𝑘 so that 2^𝑘−1≤𝑛≤2^𝑘. Then, 𝜆(2^𝑘−1)≤𝜆(𝑛)≤𝜆(2^𝑘), (2.21) and so 𝑀⁻²2^−1/𝑞𝑛^1/𝑞≤𝜆(𝑛)≤𝑀²2^1/𝑞𝑛^1/𝑞. (2.22) If 𝐴 is any finite subset of ℕ, by (2.3) we have 𝑀⁻¹𝜆(|𝐴|)≤‖_𝑗∈𝐴𝑥_𝑗‖≤𝑀𝜆(|𝐴|), (2.23) hence 𝐶⁻¹|𝐴|^1/𝑞≤‖_𝑗∈𝐴𝑥_𝑗‖≤𝐶|𝐴|^1/𝑞, (2.24) where 𝐶=𝑀³2^1/𝑞.
To prove the equivalence of (𝑥_𝑖)^∞_𝑖=1 with the canonical basis of ℓ_𝑞, given any 𝑛∈ℕ we let (𝑎_𝑖)^𝑛_𝑖=1 be nonnegative scalars such that 𝑎^𝑞_𝑖∈ℚ and ∑^𝑛_𝑖=1𝑎^𝑞_𝑖=1. Each 𝑎^𝑞_𝑖 can be written in the form 𝑎^𝑞_𝑖=𝑚_𝑖/𝑚 where 𝑚_𝑖∈ℕ, 𝑚 is de common denominator of the 𝑎^𝑞_𝑖's, and ∑^𝑛_𝑖=1𝑚_𝑖=𝑚.
Let 𝐴₁ be interval of natural numbers [1,𝑚₁] and for 𝑗=2,…,𝑛 let 𝐴_𝑖 be the interval of natural numbers [𝑚₁+⋯+𝑚_𝑖−1+1,𝑚₁+⋯+𝑚_𝑖]. The sets 𝐴₁,…,𝐴_𝑛 are disjoint and have cardinality |𝐴_𝑖|=𝑚_𝑖 for each 𝑖=1,…,𝑛. Consider the normalized constant coefficient block basic sequence defined as 𝑢_𝑖=𝑐_𝑖⁻¹_{𝑗∈𝐴𝑖}𝑥_𝑗,1≤𝑖≤𝑛, (2.25) where 𝑐_𝑖=‖∑_{𝑗∈𝐴𝑖}𝑥_𝑘‖. Equation (2.24) yields 𝐶⁻¹𝑚_𝑖^1/𝑞≤𝑐_𝑖≤𝐶𝑚_𝑖^1/𝑞,1≤𝑖≤𝑛. (2.26) Therefore, 𝐶⁻¹𝑚^1/𝑞‖^𝑛_{𝑖=1𝑗∈𝐴𝑖}𝑥_𝑗‖≤‖^𝑛_𝑖=1𝑎_𝑖𝑢_𝑖𝐶‖≤𝑚^1/𝑞‖^𝑛_{𝑖=1𝑗∈𝐴𝑖}𝑥_𝑘‖, (2.27) that is, 𝐶⁻¹𝜆(𝑚)𝑚^1/𝑞≤‖^𝑛_𝑖=1𝑎_𝑖𝑢_𝑖‖≤𝐶𝜆(𝑚)𝑚^1/𝑞. (2.28) Thus, 1𝐶²𝑀≤‖^𝑛_𝑖=1𝑎_𝑖𝑢_𝑖‖≤𝐶²𝑀. (2.29) Using (2.2) again, we have 1𝐶²𝑀²≤‖^𝑛_𝑖=1𝑎_𝑖𝑥_𝑖‖≤𝐶²𝑀². (2.30) We note that a simple density argument shows that (2.30) holds whenever ∑^𝑛_𝑖=1|𝑎_𝑖|^𝑞=1 (i.e., without the assumption that |𝑎_𝑖|^𝑞 is rational), and this completes the proof that (𝑥_𝑖)^∞_𝑖=1 is equivalent to the canonical ℓ_𝑞-basis for some 𝑝≤𝑞<∞. Since 𝑋 is not locally convex, we conclude that 𝑝≤𝑞<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.