Equivalences involving (p,q)-multi-norms
We consider (p,q)-multi-norms and standard t-multi-norms based on Banach spaces of the form , and resolve some question about the mutual equivalence of two such multi-norms. We introduce a new multi-norm, called the [p,q]-concave multi-norm, and relate it to the standard t-multi-norm.
Erratum to the paper: Ideals of homogeneous polynomials and weakly compact approximation property in Banach spaces published in Czech. Math. J. vol. 57 (132), No. 2 (2007), 763–776
Extension and lifting of weakly continuous polynomials
We show that a Banach space X is an ℒ₁-space (respectively, an -space) if and only if it has the lifting (respectively, the extension) property for polynomials which are weakly continuous on bounded sets. We also prove that X is an ℒ₁-space if and only if the space of m-homogeneous scalar-valued polynomials on X which are weakly continuous on bounded sets is an -space.
Extension of bilinear forms from subspaces of -spaces.
Extension of multilinear operators on Banach spaces.
These notes deal with the extension of multilinear operators on Banach spaces. The organization of the paper is as follows. In the first section we study the extension of the product on a Banach algebra to the bidual and some related structures including modules and derivations. Tha approach is elementary and uses the classical Arens' technique. Actually most of the results of section 1 can be easily derived from section 2. In section 2 we consider the problem of extending multilinear forms on a...
Extreme points in tensor products and a theorem of de Leeuw
Finite rank approximation and semidiscreteness for linear operators
Given a completely bounded map from an operator space into a von Neumann algebra (or merely a unital dual algebra) , we define to be -semidiscrete if for any operator algebra , the tensor operator is bounded from into , with norm less than . We investigate this property and characterize it by suitable approximation properties, thus generalizing the Choi-Effros characterization of semidiscrete von Neumann algebras. Our work is an extension of some recent work of Pisier on an analogous...
Formes bilinéaires hankéliennes compactes sur H2 (X) X H2 (Y).
Ideals of finite rank operators, intersection properties of balls, and the approximation property
We characterize the approximation property of Banach spaces and their dual spaces by the position of finite rank operators in the space of compact operators. In particular, we show that a Banach space E has the approximation property if and only if for all closed subspaces F of , the space ℱ(F,E) of finite rank operators from F to E has the n-intersection property in the corresponding space K(F,E) of compact operators for all n, or equivalently, ℱ(F,E) is an ideal in K(F,E).
Ideals of homogeneous polynomials and weakly compact approximation property in Banach spaces
We show that a Banach space has the weakly compact approximation property if and only if each continuous Banach-valued polynomial on can be uniformly approximated on compact sets by homogeneous polynomials which are members of the ideal of homogeneous polynomials generated by weakly compact linear operators. An analogous result is established also for the compact approximation property.
Inner M-ideals in Banach algebras.
Integral multilinear forms on spaces
We use polymeasures to characterize when a multilinear form defined on a product of spaces is integral.
Integral operators on the section space of a Banach bundle.
Isomorphic properties in spaces of compact operators
We introduce the definition of -limited completely continuous operators, . The question of whether a space of operators has the property that every -limited subset is relative compact when the dual of the domain and the codomain have this property is studied using -limited completely continuous evaluation operators.
Johnson's projection, Kalton's property (M*), and M-ideals of compact operators
Let X and Y be Banach spaces. We give a “non-separable” proof of the Kalton-Werner-Lima-Oja theorem that the subspace (X,X) of compact operators forms an M-ideal in the space (X,X) of all continuous linear operators from X to X if and only if X has Kalton’s property (M*) and the metric compact approximation property. Our proof is a quick consequence of two main results. First, we describe how Johnson’s projection P on (X,Y)* applies to f ∈ (X,Y)* when f is represented via a Borel (with respect to...
-convexity and duality for almost summing operators.
La structure des sous-espaces de treillis [Book]
Lifting of certain isomorphic properties to .
Linear operations tensor products, and contractive projections in function spaces