Techniques probabilistes pour l'étude de problèmes d'isomorphismes entre espaces de Banach
The Banach space H...(X, d, ...).II.
The Banach space S is complementably minimal and subsequentially prime
We first include a result of the second author showing that the Banach space S is complementably minimal. We then show that every block sequence of the unit vector basis of S has a subsequence which spans a space isomorphic to its square. By the Pełczyński decomposition method it follows that every basic sequence in S which spans a space complemented in S has a subsequence which spans a space isomorphic to S (i.e. S is a subsequentially prime space).
The classical subspaces of the projective tensor products of and C(α) spaces, α < ω₁
We completely determine the and C(K) spaces which are isomorphic to a subspace of , the projective tensor product of the classical space, 1 ≤ p < ∞, and the space C(α) of all scalar valued continuous functions defined on the interval of ordinal numbers [1,α], α < ω₁. In order to do this, we extend a result of A. Tong concerning diagonal block matrices representing operators from to ℓ₁, 1 ≤ p < ∞. The first main theorem is an extension of a result of E. Oja and states that the only...
The cofinal property of the reflexive indecomposable Banach spaces
It is shown that every separable reflexive Banach space is a quotient of a reflexive hereditarily indecomposable space, which yields that every separable reflexive Banach is isomorphic to a subspace of a reflexive indecomposable space. Furthermore, every separable reflexive Banach space is a quotient of a reflexive complementably -saturated space with and of a saturated space.
The complemented subspace problem revisited
We show that if X is an infinite-dimensional Banach space in which every finite-dimensional subspace is λ-complemented with λ ≤ 2 then X is (1 + C√(λ-1))-isomorphic to a Hilbert space, where C is an absolute constant; this estimate (up to the constant C) is best possible. This answers a question of Kadets and Mityagin from 1973. We also investigate the finite-dimensional versions of the theorem.
The Distortion of Hilbert Space.
The dual form of the approximation property for a Banach space and a subspace
Given a Banach space X and a subspace Y, the pair (X,Y) is said to have the approximation property (AP) provided there is a net of finite rank bounded linear operators on X all of which leave the subspace Y invariant such that the net converges uniformly on compact subsets of X to the identity operator. In particular, if the pair (X,Y) has the AP then X, Y, and the quotient space X/Y have the classical Grothendieck AP. The main result is an easy to apply dual formulation of this property. Applications...
The Dual of a Non-reflexive L-embedded Banach Space Contains Isometrically
A Banach space is said to be L-embedded if it is complemented in its bidual in such a way that the norm between the two complementary subspaces is additive. We prove that the dual of a non-reflexive L-embedded Banach space contains isometrically.
The Dunford-Pettis property, the Gelfand-Phillips property, and L-sets
The Dunford-Pettis property and the Gelfand-Phillips property are studied in the context of spaces of operators. The idea of L-sets is used to give a dual characterization of the Dunford-Pettis property.
The fixed-point property in Banach spaces containing a copy of .
The functor σ²X
We disprove the existence of a universal object in several classes of spaces including the class of weakly Lindelöf Banach spaces.
The Gruenhage property, property *, fragmentability, and σ-isolated networks in generalized ordered spaces
We examine the Gruenhage property, property * (introduced by Orihuela, Smith, and Troyanski), fragmentability, and the existence of σ-isolated networks in the context of linearly ordered topological spaces (LOTS), generalized ordered spaces (GO-spaces), and monotonically normal spaces. We show that any monotonically normal space with property * or with a σ-isolated network must be hereditarily paracompact, so that property * and the Gruenhage property are equivalent in monotonically normal spaces....
The heredity problem for weakly compactly generated Banach spaces
The Maurey extension property for Banach spaces with the Gordon-Lewis property and related structures
The main result of this paper states that if a Banach space X has the property that every bounded operator from an arbitrary subspace of X into an arbitrary Banach space of cotype 2 extends to a bounded operator on X, then every operator from X to an L₁-space factors through a Hilbert space, or equivalently . If in addition X has the Gaussian average property, then it is of type 2. This implies that the same conclusion holds if X has the Gordon-Lewis property (in particular X could be a Banach...
The maximum distance between two-dimensional Banach spaces.
The problem of complementability for some spaces of vector measures of bounded variation with values in Banach spaces containing copies of
Let (S, ∑, m) be any atomless finite measure space, and X any Banach space containing a copy of . Then the Bochner space is uncomplemented in ccabv(∑,m;X), the Banach space of all m-continuous vector measures that are of bounded variation and have a relatively compact range; and ccabv(∑,m;X) is uncomplemented in cabv(∑,m;X). It is conjectured that this should generalize to all Banach spaces X without the Radon-Nikodym property.
The Schroeder-Bernstein index for Banach spaces
In relation to some Banach spaces recently constructed by W. T. Gowers and B. Maurey, we introduce the notion of Schroeder-Bernstein index SBi(X) for every Banach space X. This index is related to complemented subspaces of X which contain some complemented copy of X. Then we establish the existence of a Banach space E which is not isomorphic to Eⁿ for every n ∈ ℕ, n ≥ 2, but has a complemented subspace isomorphic to E². In particular, SBi(E) is uncountable. The construction of E follows the approach...
The set of exponents for isomorphic extensions of Hölder maps is not always closed.
The space of compact operators contains when a noncompact operator is suitably factorized