Unit Vector Basis Research Articles

Banach spaces with the Lebesgue property of Riemann integrability

A Banach space is said to have the Lebesgue property if every Riemann-integrable function f:[0,1]→X is Lebesgue almost everywhere continuous. We give a characterization of the Lebesgue property in terms of a new sequential asymptotic structure that is strictly between the notions of spreading and asymptotic models. We also reproduce an apparently lost theorem of Pelczynski and da Rocha Filho that a subspace X⊂L1[0,1] has the Lebesgue property if every spreading model of X is equivalent to the unit vector basis of ℓ1.

New characterizations of the unit vector basis of or

AbstractMotivated by Altshuler’s famous characterization of the unit vector basis of $c_0$ or $\ell _p$ among symmetric bases (Altshuler [1976, Israel Journal of Mathematics, 24, 39–44]), we obtain similar characterizations among democratic bases and among bidemocratic bases. We also prove a separate characterization of the unit vector basis of $\ell _1$ .

Disjointly homogeneous Orlicz spaces revisited

Let \(1 \le p \le \infty \). A Banach lattice X is said to be p-disjointly homogeneous or \((p-DH)\) (resp. restricted \((p-DH)\)) if every normalized disjoint sequence in X (resp. every normalized sequence of characteristic functions of disjoint subsets) contains a subsequence equivalent in X to the unit vector basis of \(\ell _p\). We revisit DH-properties of Orlicz spaces and refine some previous results of this topic, showing that the \((p-DH)\)-property is not stable under duality in the class of Orlicz spaces and the classes of restricted \((p-DH)\) and \((p-DH)\) Orlicz spaces are different. Moreover, we give a characterization of uniform \((p-DH)\) Orlicz spaces and establish also closed connections between this property and the duality of the DH-property.

Closed ideals of operators on the Tsirelson and Schreier spaces

Let B(X) denote the Banach algebra of bounded operators on X, where X is either Tsirelson's Banach space or the Schreier space of order n for some n∈N. We show that the lattice of closed ideals of B(X) has a very rich structure; in particular B(X) contains at least continuum many maximal ideals.Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection PN∈B(X) corresponding to each non-empty subset N of N. A closed ideal of B(X) is spatial if it is generated by PN for some N. We can now state our main conclusions as follows:•the family of spatial ideals lying strictly between the ideal of compact operators and B(X) is non-empty and has no minimal or maximal elements;•for each pair I⫋J of spatial ideals, there is a family {ΓL:L∈Δ}, where the index set Δ has the cardinality of the continuum, such that ΓL is an uncountable chain of spatial ideals, ⋃ΓL is a closed ideal that is not spatial, andI⫋L⫋JandL+M‾=J whenever L,M∈Δ are distinct and L∈ΓL, M∈ΓM.

On the complete separation of asymptotic structures in Banach spaces

Let (ei)i denote the unit vector basis of ℓp, 1≤p<∞, or c0. We construct a reflexive Banach space with an unconditional basis that admits (ei)i as a uniformly unique spreading model while it has no subspace with a unique asymptotic model, and hence it has no asymptotic-ℓp or c0 subspace. This solves a problem of E. Odell. We also construct a space with a unique ℓ1 spreading model and no subspace with a uniformly unique ℓ1 spreading model. These results are achieved with the utilization of a new version of the method of saturation under constraints that uses sequences of functionals with increasing weights.

Some remarks about disjointly homogeneous symmetric spaces

Let $1\le p<\infty$. A symmetric space $X$ on $[0,1]$ is said to be $p$-disjointly homogeneous (resp. restricted $p$-disjointly homogeneous) if every sequence of normalized pairwise disjoint functions from $X$ (resp. characteristic functions) contains a subsequence equivalent in $X$ to the unit vector basis of $l_p$. Answering a question posed recently, we construct, for each $1\le p<\infty$, a restricted $p$-disjointly homogeneous symmetric space, which is not $p$-disjointly homogeneous. Moreover, we prove that the property of $p$-disjoint homogeneity is preserved under Banach isomorphisms.

On the existence of a basis in a complemented subspace of a nuclear Köthe space from class

A proof is presented that an arbitrary complemented subspace of a Köthe nuclear space from class has a basis, provided that the relevant Köthe matrix is regular in the sense of Dragilev. It is also shown that each such subspace must have a basis that is quasi-equivalent to a part of the canonical unit-vector basis. Bibliography: 21 titles.

Duality problem for disjointly homogeneous rearrangement invariant spaces

Let 1≤p<∞. A Banach lattice E is said to be disjointly homogeneous (resp. p-disjointly homogeneous) if two arbitrary normalized disjoint sequences from E contain equivalent in E subsequences (resp. every normalized disjoint sequence contains a subsequence equivalent in E to the unit vector basis of lp). Answering a question raised in the paper [11], for each 1<p<∞, we construct a reflexive p-disjointly homogeneous rearrangement invariant space on [0,1] whose dual is not disjointly homogeneous. Employing methods from interpolation theory, we provide new examples of disjointly homogeneous rearrangement invariant spaces; in particular, we show that there is a Tsirelson type disjointly homogeneous rearrangement invariant space, which contains no subspace isomorphic to lp, 1≤p<∞, or c0.

On spreading sequences and asymptotic structures

In the first part of the paper we study the structure of Banach spaces with a conditional spreading basis. The geometry of such spaces exhibits a striking resemblance to the geometry of James space. Further, we show that the averaging projections onto subspaces spanned by constant coefficient blocks with no gaps between supports are bounded. As a consequence, every Banach space with a spreading basis contains a complemented subspace with an unconditional basis. This gives an affirmative answer to a question of H. Rosenthal. The second part contains two results on Banach spaces X X whose asymptotic structures are closely related to c 0 c_0 and do not contain a copy of ℓ 1 \ell _1 : i) Suppose X X has a normalized weakly null basis ( x i ) (x_i) and every spreading model ( e i ) (e_i) of a normalized weakly null block basis satisfies ‖ e 1 − e 2 ‖ = 1 \|e_1-e_2\|=1 . Then some subsequence of ( x i ) (x_i) is equivalent to the unit vector basis of c 0 c_0 . This generalizes a similar theorem of Odell and Schlumprecht and yields a new proof of the Elton–Odell theorem on the existence of infinite ( 1 + ε ) (1+\varepsilon ) -separated sequences in the unit sphere of an arbitrary infinite dimensional Banach space. ii) Suppose that all asymptotic models of X X generated by weakly null arrays are equivalent to the unit vector basis of c 0 c_0 . Then X ∗ X^* is separable and X X is asymptotic- c 0 c_0 with respect to a shrinking basis ( y i ) (y_i) of Y ⊇ X Y\supseteq X .

1-greedy renormings of Garling sequence spaces

We show that all Garling sequence spaces admit a renorming with respect to which their standard unit vector basis is 1-greedy. We also discuss some additional properties of these Banach spaces related to uniform convexity and superreflexivity. In particular, our approach to the study of the superreflexivity of Garling sequence spaces provides an example of how essentially non-linear tools from greedy approximation can be used to shed light on the linear structure of these spaces.

ON EQUIVALENCE RELATIONS GENERATED BY SCHAUDER BASES

AbstractIn this article, a notion of Schauder equivalence relation ℝℕ/L is introduced, where L is a linear subspace of ℝℕ and the unit vectors form a Schauder basis of L. The main theorem is to show that the following conditions are equivalent:(1) the unit vector basis is boundedly complete;(2) L is a Fσ in ℝℕ;(3) ℝℕ/L is Borel reducible to ℓ∞.We show that any Schauder equivalence relation generalized by a basis of ℓ2 is Borel bireducible to ℝℕ/ℓ2 itself, but it is not true for bases of c0 or ℓ1. Furthermore, among all Schauder equivalence relations generated by sequences in c0, we find the minimum and the maximum elements with respect to Borel reducibility.

A new approach to low-distortion embeddings of finite metric spaces into non-superreflexive Banach spaces

The main goal of this paper is to develop a new embedding method which we use to show that some finite metric spaces admit low-distortion embeddings into all non-superreflexive spaces. This method is based on the theory of equal-signs-additive sequences developed by Brunel and Sucheston (1975–1976). We also show that some of the low-distortion embeddability results obtained using this method cannot be obtained using the method based on the factorization between the summing basis and the unit vector basis of ℓ1, which was used by Bourgain (1986) and Johnson and Schechtman (2009).

Rademacher functions in Nakano spaces

The closed span of Rademacher functions is investigated in Nakano spaces Lp(⋅) on [0,1] equipped with the Lebesgue measure. The main result of this paper states that under some conditions on distribution of the exponent function p the Rademacher functions form in Lp(⋅) a basic sequence equivalent to the unit vector basis in l2.

Lorentz Spaces and Embeddings Induced by Almost Greedy Bases in Banach Spaces

The aim of this paper is to undertake a systematic qualitative study of the built-in symmetry of almost greedy bases in Banach spaces. More specifically, by refining the techniques that Wojtaszczyk used in J Approx Theory 107(2), 293–314 2000 for quasi-greedy bases in Hilbert spaces, we show that an almost greedy basis in a Banach space X naturally induces embeddings that allow sandwiching X between two symmetric sequence spaces. Using classical interpolation techniques in combination with duality, we also explore what we label as interpolation of greedy bases. It is then proved that the only almost greedy basis shared by any two \(\ell _{p}\) spaces is equivalent to the standard unit vector basis and that there is no basis which is simultaneously (normalized and) greedy in two different \(L_{p}\) spaces. As a by-product of our work, we obtain a new characterization of greedy bases in Banach spaces in terms of bounded linear operators.

On an isomorphic Banach–Mazur rotation problem and maximal norms in Banach spaces

We prove that the spaces ℓp, 1<p<∞, p≠2, and all infinite-dimensional subspaces of their quotient spaces do not admit equivalent almost transitive renormings. This is a step towards the solution of the Banach–Mazur rotation problem, which asks whether a separable Banach space with a transitive norm has to be isometric or isomorphic to a Hilbert space. We obtain this as a consequence of a new property of almost transitive spaces with a Schauder basis, namely we prove that in such spaces the unit vector basis of ℓ22 belongs to the two-dimensional asymptotic structure and we obtain some information about the asymptotic structure in higher dimensions.Further, we prove that the spaces ℓp, 1<p<∞, p≠2, have continuum different renormings with 1-unconditional bases each with a different maximal isometry group, and that every symmetric space other than ℓ2 has at least a countable number of such renormings. On the other hand we show that the spaces ℓp, 1<p<∞, p≠2, have continuum different renormings each with an isometry group which is not contained in any maximal bounded subgroup of the group of isomorphisms of ℓp.

Renorming spaces with greedy bases

We study the problem of improving the greedy constant or the democracy constant of a basis of a Banach space by renorming. We prove that every Banach space with a greedy basis can be renormed, for a given ε>0, so that the basis becomes (1+ε)-democratic, and hence (2+ε)-greedy, with respect to the new norm. If in addition the basis is bidemocratic, then there is a renorming so that in the new norm the basis is (1+ε)-greedy. We also prove that in the latter result the additional assumption of the basis being bidemocratic can be removed for a large class of bases. Applications include the Haar systems in Lp[0,1], 1<p<∞, and in dyadic Hardy space H1, as well as the unit vector basis of Tsirelson space.

Disjointly homogeneous rearrangement invariant spaces via interpolation

A Banach lattice E is called p-disjointly homogeneous, 1≤p≤∞, when every sequence of pairwise disjoint normalized elements in E has a subsequence equivalent to the unit vector basis of ℓp. Employing methods from interpolation theory, we clarify which r.i. spaces on [0,1] are p-disjointly homogeneous. In particular, for every 1<p<∞ and any increasing concave function φ on [0,1], which is not equivalent to neither 1 nor t, there exists a p-disjointly homogeneous r.i. space with the fundamental function φ. Moreover, it is shown that given 1<p<∞ and an increasing concave function φ with non-trivial dilation indices, there is a unique p-disjointly homogeneous space among all interpolation spaces between the Lorentz and Marcinkiewicz spaces associated with φ.

Rearrangement invariant spaces with Kato property

We study rearrangement invariant spaces on which the classes of strictly singular and compact operators coincide. The relation between this property and the fact that every normalized disjoint sequence in the space has a subsequence equivalent to the unit vector basis of $\ell_2$ is analyzed.

Extreme points of the set of Banach limits

Let \({\mathbb{N}}\) be the set of all natural numbers and \({\ell_\infty=\ell_\infty (\mathbb{N})}\) be the Banach space of all bounded sequences x = (x1, x2 . . .) with the norm $$\|x\|_{\infty}=\sup_{n\in\mathbb{N}}|x_n|,$$ and let \({\ell_\infty^*}\) be its Banach dual. Let \({\mathfrak{B} \subset \ell_\infty^*}\) be the set of all normalised positive translation invariant functionals (Banach limits) on l∞ and let \({ext(\mathfrak{B})}\) be the set of all extreme points of \({\mathfrak{B}}\) . We prove that an arbitrary sequence (Bj)j ≥ 1, of distinct points from the set \({ext(\mathfrak{B})}\) is 1-equivalent to the unit vector basis of the space l1 of all summable sequences. We also study Cesaro-invariant Banach limits. In particular, we prove that the norm closed convex hull of \({ext(\mathfrak{B})}\) does not contain a Cesaro-invariant Banach limit.

Optimally Sparse Frames

Frames have established themselves as a means to derive redundant, yet stable decompositions of a signal for analysis or transmission, while also promoting sparse expansions. However, when the signal dimension is large, the computation of the frame measurements of a signal typically requires a large number of additions and multiplications, and this makes a frame decomposition intractable in applications with limited computing budget. To address this problem, in this paper, we focus on frames in finite-dimensional Hilbert spaces and introduce sparsity for such frames as a new paradigm. In our terminology, a sparse frame is a frame whose elements have a sparse representation in an orthonormal basis, thereby enabling low-complexity frame decompositions. To introduce a precise meaning of optimality, we take the sum of the numbers of vectors needed from this orthonormal basis when expanding each frame vector as sparsity measure. We then analyze the recently introduced algorithm Spectral Tetris for construction of unit norm tight frames and prove that the tight frames generated by this algorithm are in fact optimally sparse with respect to the standard unit vector basis. Finally, we show that even the generalization of Spectral Tetris for the construction of unit norm frames associated with a given frame operator produces optimally sparse frames.