A sequentially unconditional Banach space with few operators

Unconditional Basis and Gordon–Lewis Constants for Spaces of Polynomials

New examples of [formula omitted]-saturated Banach spaces II

A Banach space X is prime if every infinite-dimensional complemented subspace contains a further subspace which is isomorphic to X. A Banach space X is said to be primary if whenever X — Y © Z, X is isomorphic to either Y or Z. The classical examples of prime spaces are the spaces ipj 1 < p < oc. Many spaces derived from the £p-spaces in various ways are primary (see for example [AEO] and [CL]). The primarity of B(H) was shown by Blower [B] in 1990, and Arias [A] has recently developed further techniques which are used to prove the primarity of Cι, the space of trace class operators (this was first shown by Arazy [Arl, Ar2]). It has become clear that these techniques are not naturally confined to a Hubert space context; in the present paper we wish to extend the results to a variety of tensor products and operator spaces of ^-spaces (and in some cases £p-spaces). We also include some related results. Some of the intermediate propositions (on factoring operators through the identity) may actually be true for a wider class of Banach spaces (those with unconditional bases which have nontrivial lower and upper estimates). In fact, the combinatorial aspects of the factorization can be applied quite generally, and may have other applications. The proofs of primarity, however, rely on Pelczyήski's decomposition method which is not so readily extended. We have thus kept mainly to the case of injective and projective tensor products of tv spaces throughout. The results we obtain apply to the growing study of polynomials on Banach spaces since polynomials may be considered as symmetric multilinear operators with an equivalent norm (see [FJ], [M],

On constructing multiplications on Banach spaces

For each λ ≥ 1 a class of Banach spaces φλ is defined. Isometric results are obtained on the equivalence between a measure of non-compactness and the essential norm of a linear operator defined on a φλ space. Best values of λ for the classical Banach spaces and for spaces with unconditional basis are investigated. For the space c of convergent sequences the non-existence of a λ-unconditional basis with λ &lt; 2 is deduced.

The strong and uniform ball-covering properties

Corrigendum to “Approximation by [formula omitted]-smooth, Lipschitz functions on Banach spaces” [J. Math. Anal. Appl. 315 (2006) 599–605

We are interested in the question when a Banach space X X with an unconditional basis is isomorphic (as a Banach space) to an order-continuous nonatomic Banach lattice. We show that this is the case if and only if X X is isomorphic as a Banach space with X ( ℓ 2 ) X({\ell _2}) . This and results of Bourgain are used to show that spaces H 1 ( T n ) {H_1}({{\mathbf {T}}^n}) are not isomorphic to nonatomic Banach lattices. We also show that tent spaces introduced by Coifman, Meyer, and Stein are isomorphic to Rad H 1 \operatorname {Rad} \;{H_1} .

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.

Every Banach space with a non-shrinking (unconditional) basis (Xi) can be renormed so that the biorthogonal sequence has a much smaller (unconditional) basis constant than (xi). On the other hand, if the unconditional constant of is C &lt; 2 then the unconditional constant of (xi) is at most C/(2—C). This estimate is sharp.

Weak ∗-Polish Banach spaces

Quasi-greedy bases in ℓp (0 < p < 1) are democratic

Espaces de Banach ayant la propriété de Daugavet

A class of hereditarily $\ell_{p}$ ($1 \leq p \lt \infty$) Banach sequence spaces is constructed and denoted by $X_{\alpha,p}$. Any constructed space is a dual space. We show that (i) the predual of any member $X$ of the class of $X_{\alpha,1}$ contains asymptotically isometric copies of $c_0$.(ii) Every infinite dimensional subspace of $X$ contains asymptotically isometric complemented copies of $\ell_{1}$, and consequently, the dual X$^*$ of X contains subspaces isometrically isomorphic to $C[0,1]^*$. (iii) Every member of the class of $X_{\alpha,p}$ ($1 \leq p \lt \infty$) fails the Dunford-Pettis property. (iv) We observe that all $X_{\alpha,p}$ spaces are Banach spaces without unconditional basis but all constructed spaces contain a subspace which is weakly sequentially complete with an unconditional basis which is weakly null sequence but not in norm. (v) All spaces have asymptotic-norming and Kadec-Klee property. The predual of any $X_{\alpha,p}$ is an Asplund space.

A classical result due to M. Eidelheit and B. Yood states that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm on the same algebra. This definition does not assume that the arbitrary algebra norm is complete. In cases when the standard algebra norm is, in addition, maximal, it is therefore unique up to equivalence. More recently, M. Meyer showed that the Calkin algebras of a very restricted class of Banach spaces also have unique algebra norms. We generalise the Eidelheit-Yood method of proof, to show that the conventional quotient norm on a larger class of Calkin algebras is minimal. Since maximality of the norm is a presumed property for the class, the norm is also unique. We thus extend the result of Meyer. In particular, we establish that the Calkin algebras of canonical Banach spaces such as James’ space and Tsirelson’s space have unique algebra norms, without assuming completeness. We also prove uniqueness of norm for quotients of the algebras of operators on classical non-separable spaces, the closed ideals of which were previously studied by M. Daws. One aspect of the Eidelheit-Yood method is a dependence on the uniform boundedness principle. As a component of our generalisation, we prove an analogue of that principle which applies to Calkin algebra elements rather than bounded linear operators. In order to translate the uniform boundedness principle into this new setting, we take the perspective that non-compact operators map certain wellseparated sequences to other well-separated sequences. We analyse the limiting separation of such sequences, using these values to measure the non-compactness of operators and define the requisite notion of a bounded set of non-compact operators. In the cases when the underlying Banach space has a Schauder basis, we are able to restrict attention to seminormalised block basic sequences. As a consequence, our main uniqueness of norm result for Calkin algebras relies on the existence of bounded mappings between, and projections onto, the spans of block basic sequences in the relevant Banach spaces.