Uniqueness of unconditional basis in quasi-Banach spaces which are not sufficiently Euclidean

Strongly absolute bases are, roughly speaking, purely nonlocally convex bases in quasi-Banach spaces. When, in addition, they are unconditional then the discrete lattice structure they induce in the space is lattice anti-Euclidean. In this brief note we characterize the complemented unconditional basic sequences in those quasi-Banach spaces with strongly absolute unconditional basis, and use this result to derive the uniqueness of unconditional basis in many classical quasi-Banach spaces.     

Remarks on the Existence of Unconditional Bases for Weighted Countably-Hilbert Spaces and Their Complemented Subspaces

We give versions of a criterion for existence of unconditional bases for countably-Hilbert spaces. As applications we obtain theorems on existence of unconditional bases for certain classes of countably-Hilbert function spaces and for their complemented subspaces under additional constraints on the space and the corresponding projections to the complemented subspaces. These classes include generalizations of power series spaces of finite type and Kothe spaces determined by Dragilev-type functions.     

An alternative approach to the uniqueness of unconditional basis of for

We give an alternative and much simpler proof of the uniqueness of unconditional basis (up to equivalence and permutation) in the quasi-Banach spaces ℓ_p(c₀) for 0<p<1 and its complemented subspaces with unconditional basis. The new approach uses the fact that the Banach envelope of these spaces is not sufficiently Euclidean with the lattice structure induced by its unconditional basis.     

Ergodic Banach spaces

We show that any Banach space contains a continuum of non-isomorphic subspaces or a minimal subspace. We define an ergodic Banach space X as a space such that E₀ Borel reduces to isomorphism on the set of subspaces of X, and show that every Banach space is either ergodic or contains a subspace with an unconditional basis which is complementably universal for the family of its block-subspaces. We also use our methods to get uniformity results. We show that an unconditional basis of a Banach space, of which every block-subspace is complemented, must be asymptotically c₀ or ?_p, and we deduce some new characterisations of the classical spaces c₀ and ?_p.     

Quasi-minimality and tightness by range in spaces with unconditional basis

We present a reflexive Banach space with an unconditional basis which is quasi-minimal and tight by range, i.e., of type (4) in the Ferenczi-Rosendal list within the framework of Gowers’ classification program of Banach spaces. The space is an unconditional variant of the Gowers Hereditarily Indecomposable space with an asymptotically unconditional basis.     

A remark on symmetric bases

Every separable Banach space with an unconditional basis is isomorphic to a complemented subspace of a space with a symmetric basis.     

Embedding spaces with unconditional bases

Every super-reflexive space with an unconditional basis is isomorphic to a complemented subspace of a super-reflexive space with a symmetric basis.     

Embedding Banach spaces with unconditional bases into spaces with symmetric bases

Every reflexive Banach space with unconditional basis is isomorphic to a complemented subspace of a reflexive Banach space with symmetric basis.     

关于Banach空间的无条件基

给出具有唯一无条件基的无穷维Banach空间,并给出其无条件基的若干性质.     

Discrete time analytic semigroups and the geometry of Banach spaces

We study different notions of discrete maximal regularity for discrete-time abstract Cauchy problems in Banach spaces. First we look at l ₂-discrete maximal regularity and show that Hilbert spaces are the only Banach spaces, among spaces with an unconditional basis, in which the analyticity of the associated discrete-time semigroup is a sufficient condition to obtain this kind of regularity. We then turn to different notions of regularity, in a l ₁ and in a l _∞ sense. We link the existence of particular semigroups such that the associated Cauchy problem has one of these maximal regularities to the geometry of the underlying Banach space (more precisely, to the existence of a complemented subspace isomorphic to c ₀ or l ₁). Finally, we give some elements to compare these regularities.     

Uniqueness of unconditional bases in nonlocally convex c<Subscript>0</Subscript>-products

We show that the c ₀-product (X ⊕ X ⊕ ... ⊕ X ⊕ ...)₀ of a natural quasi-Banach space X with strongly absolute unconditional basis has a unique unconditional basis up to permutation. Our results apply to a wide range of cases, including most of the c ₀-products of the nonlocally convex classical quasi-Banach spaces.     

Uniqueness of unconditional bases in quasi-banach spaces with applications to hardy spaces, II

We prove that a wide class of quasi-Banach spaces has a unique up to a permutation unconditional basis. This applies in particular to Hardy spacesH _p forp<1. We also investigate the structure of complemented subspaces ofH _p (D). The proofs use in essential way matching theory. This research was supported in part by KBN grant N. 2P301004.06 and by the Overseas Visiting Scholarship of St. John’s College, Cambridge.     

A remark on unconditional basic sequences inL p (1<p<∞)

We prove that every separable ? _p space (1<p<∞), with an unconditional basis is isomorphic to a complemented subspace ofL _p which is spanned by a block basis of the Haar system.     

Frequent hypercyclicity,chaos, and unconditional Schauder decompositions

We prove that if X is any complex separable infinite-dimensional Banach space with an unconditional Schauder decomposition, X supports an operator T which is chaotic and frequently hypercyclic. In contrast with the complex case, we observe that there are real Banach spaces with an unconditional basis which support no chaotic operator.     

Nonlinear Approximation and Muckenhoupt Weights

In the general atomic setting of an unconditional basis in a (quasi-) Banach space, we show that representing the spaces of m-terms approximation as Lorentz spaces is equivalent to the verification of two inequalities (Jackson and Bernstein), and that the validity of these two properties is equivalent to the Temlyakov property. The proof is very direct and, especially, does not use interpolation theory. We apply this result to establish a representation theorem when the norm of the (quasi-) Banach space is given by a quadratic variation formula (thanks to a condition called the p-reverse inequality). This quadratic variation framework is in fact very rich and contains, as examples, the cases of Hardy spaces. We also consider the cases of "weighted" Hardy and Lebesgue spaces when the weight belongs to a Muckenhoupt class and the basis is a wavelet basis. This provides a new example of bases well adapted to approximation.     

Discrete Time Analytic Semigroups and the Geometry of Banach Spaces

Abstract. We study different notions of discrete maximal regularity for discrete-time abstract Cauchy problems in Banach spaces. First we look at l ₂ -discrete maximal regularity and show that Hilbert spaces are the only Banach spaces, among spaces with an unconditional basis, in which the analyticity of the associated discrete-time semigroup is a sufficient condition to obtain this kind of regularity. We then turn to different notions of regularity, in a l ₁ and in a l _∞ sense. We link the existence of particular semigroups such that the associated Cauchy problem has one of these maximal regularities to the geometry of the underlying Banach space (more precisely, to the existence of a complemented subspace isomorphic to c ₀ or l ₁ ). Finally, we give some elements to compare these regularities.     

Uniqueness of unconditional bases in quasi-banach spaces with applications to Hardy spaces

We prove some general results on the uniqueness of unconditional bases in quasi-Banach spaces. We show in particular that certain Lorentz spaces have unique unconditional bases answering a question of Nawrocki and Ortynski. We then give applications of these results to Hardy spaces by showing the spacesH _p (T ⁿ ) are mutually non-isomorphic for differing values ofn when 0<p<1. The research of the first two authors was partially supported by NSF-grant DMS 8901636.     

On the Borelness of the intersection operation

We study the intersection operation of closed linear subspaces in a separable Banach space. We show that if the ambient space is quasi-reflexive, then the intersection operation is Borel. On the other hand, if the space contains a closed subspace with a Schauder decomposition into infinitely many non-reflexive spaces, then the intersection operation is not Borel. As a corollary, for a closed subspace of a Banach space with an unconditional basis, the intersection operation of the closed linear subspaces is Borel if and only if the space is reflexive. We also consider the intersection operation of additive subgroups in an infinite-dimensional separable Banach space, and show that if this intersection operation is Borel then the space is hereditarily indecomposable.     

Nuclear Fréchet spaces with locally round finite dimensional decompositions

A class of finite dimensional decompositions (FDDs), called locally round, is introduced in Fréchet spaces. A Fréchet space with a locally round FDD can be viewed as a generalization of a Köthe space. The block subspaces and block quotients of such a space are always complemented and have a basis. Conversely, sometimes these properties characterize an FDD being locally round.     

On Banach Spaces Whose Unit Sphere Determines Polynomials

In this paper we study the problem of characterizing the real Banach spaces whose unit sphere determines polynomials, i.e., if two polynomials coincide in the unit sphere, is this sufficient to guarantee that they are identical？ We show that, in the frame of spaces with unconditional basis, non- reflexivity is a sufficient, although not necessary, condition for the above question to have an affirmative answer. We prove that the only lp^n spaces having this property are those with p irrational, while the only lp spaces which do not enjoy it are those with p an even integer. We also introduce a class of polynomial determining sets in any real Banach space.