Finite Dimensional Decomposition Research Articles

Compact retractions and Schauder decompositions in Banach spaces

Let X X be a separable Banach space. We give an almost characterization of the existence of a Finite Dimensional Decomposition (FDD for short) for X X in terms of Lipschitz retractions onto generating compact subsets K K of X X . In one direction, if X X admits an FDD then we construct a Lipschitz retraction onto a small generating convex and compact set K K . On the other hand, we prove that if X X admits a “small” generating compact Lipschitz retract then X X has the π \pi -property. It is still unknown if the π \pi -property is isomorphically equivalent to the existence of an FDD. For dual Banach spaces this is true, so our results give a characterization of the FDD property for dual Banach spaces X X . We give an example of a small generating convex compact set which is not a Lipschitz retract of C [ 0 , 1 ] C[0,1] , although it is contained in a small convex Lipschitz retract and contains another one. We characterize isomorphically Hilbertian spaces as those Banach spaces X X for which every convex and compact subset is a Lipschitz retract of X X . Finally, we prove that a convex and compact set K K in any Banach space with a Uniformly Rotund in Every Direction norm is a uniform retract, of every bounded set containing it, via the nearest point map.

Banach Spaces of Almost Universal Complemented Disposition

Abstract We introduce and study the notion of space of almost universal complemented disposition (a.u.c.d.) as a generalization of Kadec space. We show that every Banach space with separable dual is isometrically contained as a $1$-complemented subspace of a separable a.u.c.d. space and that all a.u.c.d. spaces with $1$-finite dimensional decomposition (FDD) are isometric and contain isometric $1$-complemented copies of every separable Banach space with $1$-FDD. We then study spaces of universal complemented disposition (u.c.d.) and provide different constructions for such spaces. We also consider spaces of u.c.d. with respect to separable spaces.

CONCERNING SUMMABLE SZLENK INDEX

AbstractWe generalize the notion of summable Szlenk index from a Banach space to an arbitrary weak*-compact set. We prove that a weak*-compact set has summable Szlenk index if and only if its weak*-closed, absolutely convex hull does. As a consequence, we offer a new, short proof of a result from Draga and Kochanek [J. Funct. Anal. 271 (2016), 642–671] regarding the behavior of summability of the Szlenk index under c0 direct sums. We also use this result to prove that the injective tensor product of two Banach spaces has summable Szlenk index if both spaces do, which answers a question from Draga and Kochanek [Proc. Amer. Math. Soc. 145 (2017), 1685–1698]. As a final consequence of this result, we prove that a separable Banach space has summable Szlenk index if and only if it embeds into a Banach space with an asymptotic c0 finite dimensional decomposition, which generalizes a result from Odell et al. [Q. J. Math. 59, (2008), 85–122]. We also introduce an ideal norm $\mathfrak{s}$ on the class $\mathfrak{S}$ of operators with summable Szlenk index and prove that $(\mathfrak{S}, \mathfrak{s})$ is a Banach ideal. For 1 ⩽ p ⩽ ∞, we prove precise results regarding the summability of the Szlenk index of an ℓp direct sum of a collection of operators.

The scalar-plus-compact property in spaces without reflexive subspaces

A hereditarily indecomposable Banach space $\mathfrak{X}_{\mathfrak{nr}}$ is constructed that is the first known example of a $\mathscr{L}_\infty$-space not containing $c_0$, $\ell_1$, or reflexive subspaces and answers a question posed by J. Bourgain. Moreover, the space $\mathfrak{X}_{\mathfrak{nr}}$ satisfies the scalar-plus-compact property and it is the first known space without reflexive subspaces having this property. It is constructed using the Bourgain-Delbaen method in combination with a recent version of saturation under constraints in a mixed-Tsirelson setting. As a result, the space $\mathfrak{X}_{\mathfrak{nr}}$ has a shrinking finite dimensional decomposition and does not contain a boundedly complete sequence.

Dynamics of financial returns densities: A functional approach applied to the Bovespa intraday index

We model the stochastic evolution of the probability density functions (PDFs) of Ibovespa intraday returns over business days, in a functional time series framework. We find evidence that the dynamic structure of the PDFs reduces to a vector process lying in a two-dimensional space. Our main contributions are as follows. First, we provide further insights into the finite-dimensional decomposition of the curve process: it is shown that its evolution can be interpreted as a dynamic dispersion-symmetry shift. Second, we provide an application to realized volatility forecasting, with a forecasting ability that is comparable to those of HAR realized volatility models in the model confidence set framework.

Dominated operators from lattice-normed spaces to sequence Banach lattices

We show that every dominated linear operator from a Banach–Kantorovich space over an atomless Dedekind-complete vector lattice to a sequence Banach lattice ℓp(Γ) or c0(Γ) is narrow. As a consequence, we obtain that an atomless Banach lattice cannot have a finite-dimensional decomposition of a certain kind. Finally, we show that the order-narrowness of a linear dominated operator T from a lattice-normed space V to the Banach space with a mixed norm (W,F) over an order-continuous Banach lattice F implies the order-narrowness of its exact dominant |T|.

Smoothness, asymptotic smoothness and the Blum-Hanson property

We isolate various sufficient conditions for a Banach space X to have the so-called Blum-Hanson property. In particular, we show that X has the Blum-Hanson property if either the modulus of asymptotic smoothness of X has an extremal behaviour at infinity, or if X is uniformly Gâteaux smooth and embeds isometrically into a Banach space with a 1-unconditional finite-dimensional decomposition.

The finite-dimensional decomposition property in non-Archimedean Banach spaces

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property (OFDDP) if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces. This property has an influence in the non-Archimedean Grothendieck’s approximation theory, where an open problem is the following: Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E. Does D have the OFDDP? In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP. Next we prove that, however, for certain classes of Banach spaces of countable type, the OFDDP is preserved by taking finite-codimensional subspaces.

Existence and Multiplicity Results for Some Elliptic Systems in Unbounded Cylinders

We study the following nonlinear elliptic system of Lane–Emden type $$\left\{\begin{array}{ll} -\Delta u = {\rm sgn}(v) |v| ^{p-1} \qquad \qquad \qquad \; {\rm in} \; \Omega , \\ -\Delta v = - \lambda {\rm sgn} (u)|u| \frac{1}{p-1} + f(x, u)\; \; {\rm in}\; \Omega , \\ u = v = 0 \qquad \qquad \qquad \quad \quad \;\;\;\;\; {\rm on}\; \partial \Omega , \end{array}\right.$$ where \({\lambda \in \mathbb{R}}\). If \({\lambda \geq 0}\) and \({\Omega}\) is an unbounded cylinder, i.e., \({\Omega = \tilde \Omega \times \mathbb{R}^{N-m} \subset \mathbb{R}^{N}}\), \({N - m \geq 2, m \geq 1}\), existence and multiplicity results are proved by means of the Principle of Symmetric Criticality and some compact imbeddings in partially spherically symmetric spaces. We are able to state existence and multiplicity results also if \({\lambda \in \mathbb{R}}\) and \({\Omega}\) is a bounded domain in \({\mathbb{R}^{N}, N \geq 3}\). In particular, a good finite dimensional decomposition of the Banach space in which we work is given.

Infinite Asymptotic Games and (*)-Embeddings of Banach Spaces

We use methods of infinite asymptotic games to characterize subspaces of Banach spaces with a finite-dimensional decomposition (FDD) and prove new theorems on operators. We consider a separable Banach space $X$, a set $\mathcal{S}$ of sequences of fini

Subspaces and quotients of Banach spaces with shrinking unconditional bases

The main result is that a separable Banach space with the weak* unconditional tree property is isomorphic to a subspace as well as a quotient of a Banach space with a shrinking unconditional basis. A consequence of this is that a Banach space is isomorphic to a subspace of a space with a shrinking unconditional basis if and only if it is isomorphic to a quotient of a space with a shrinking unconditional basis, which solves a problem dating to the 1970s. The proof of the main result also yields that a uniformly convex space with the unconditional tree property is isomorphic to a subspace as well as a quotient of a uniformly convex space with an unconditional finite dimensional decomposition.

The Grothendieck property for injective tensor products of Banach spaces

Let X be a Banach space with the Grothendieck property, Y a reflexive Banach space, and let X ⊗ɛY be the injective tensor product of X and Y. (a) If either X** or Y has the approximation property and each continuous linear operator from X* to Y is compact, then X ⊗ɛY has the Grothendieck property. (b) In addition, if Y has an unconditional finite dimensional decomposition, then X ⊗ɛY has the Grothendieck property if and only if each continuous linear operator from X* to Y is compact.

Infinite asymptotic games

We study infinite asymptotic games in Banach spaces with a finite-dimensional decomposition (F.D.D.) and prove that analytic games are determined by characterising precisely the conditions for the players to have winning strategies. These results are applied to characterise spaces embeddable into ℓ p sums of finite dimensional spaces, extending results of Odell and Schlumprecht, and to study various notions of homogeneity of bases and Banach spaces. The results are related to questions of rapidity of subsequence extraction from normalised weakly null sequences.

Compositions of Projections in Banach Spaces and Relations Between Approximation Properties

A necessary and sucient condition for existence of a Banach space with a finite dimensional decomposition but without the -property in terms of norms of com- positions of projections is found.

Properties of the Holmes space

The following properties of the Holmes space H are established: (i) H has the Metric Approximation Property (MAP). (ii) The w ∗ -closure of the set of extreme points of the unit ball B H ∗ of the dual space H ∗ is the whole ball B H ∗ . A family of compact subsets X ⊂ U of the Urysohn space is described such that the Lipschitz-free space F ( X ) has a finite-dimensional decomposition and is not complemented in H.

A characterization of subspaces and quotients of reflexive banach spaces with unconditional bases

We prove that the dual or any quotient of a separable reflexive Banach space with the unconditional tree property (UTP) has the UTP. This is used to prove that a separable reflexive Banach space with the UTP embeds into a reflexive Banach space with an unconditional basis. This solves several longstanding open problems. In particular, it yields that a quotient of a reflexive Banach space with an unconditional finite-dimensional decomposition (UFDD) embeds into a reflexive Banach space with an unconditional basis

Some Results on the Schroeder–Bernstein Property for Separable Banach Spaces

AbstractWe construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder–Bernstein Index of any of these spaces is 2ℵ0. Our construction is based on a Banach space introduced by W. T. Gowers and B. Maurey in 1997. We also use classical descriptive set theory methods, as in some work of the first author and C. Rosendal, to improve some results of P. G. Casazza and of N. J. Kalton on the Schroeder–Bernstein Property for spaces with an unconditional finite-dimensional Schauder decomposition.

A universal reflexive space for the class of uniformly convex Banach spaces

We show that there exists a separable reflexive Banach space into which every separable uniformly convex Banach space isomorphically embeds. This solves problems raised by J. Bourgain [B] in 1980 and by W. B. Johnson in 1977 [Jo]. We also give intrinsic characterizations of separable reflexive Banach spaces which embed into a reflexive space with a block q-Hilbertian and/or a block p-Besselian finite dimensional decomposition.

Twisted sums with 𝐶(𝐾) spaces

If X X is a separable Banach space, we consider the existence of non-trivial twisted sums 0 → C ( K ) → Y → X → 0 0\to C(K)\to Y\to X\to 0 , where K = [ 0 , 1 ] K=[0,1] or ω ω . \omega ^{\omega }. For the case K = [ 0 , 1 ] K=[0,1] we show that there exists a twisted sum whose quotient map is strictly singular if and only if X X contains no copy of ℓ 1 \ell _1 . If K = ω ω K=\omega ^{\omega } we prove an analogue of a theorem of Johnson and Zippin (for K = [ 0 , 1 ] K=[0,1] ) by showing that all such twisted sums are trivial if X X is the dual of a space with summable Szlenk index (e.g., X X could be Tsirelson’s space); a converse is established under the assumption that X X has an unconditional finite-dimensional decomposition. We also give conditions for the existence of a twisted sum with C ( ω ω ) C(\omega ^{\omega }) with strictly singular quotient map.

Block diagonal polynomials

In this paper we introduce and study a certain class of polynomials in spaces with unconditional finite dimensional decomposition. Some applications to the existence of copies of ℓ ∞ \ell _\infty in spaces of polynomials and to the stabilization of polynomial algebras are given.