Infinite tensor products of Banach spaces I

In this paper we construct a new class of infinite tensor product Banach spaces. We call them spaces of type v since they are constructed in a manner which closely parallels von Neumanns construction of infinite tensor products of Hilbert spaces. We use these spaces to present results on infinite tensor products of semigroups of operators.     

On Minimal Subspaces in Tensor Representations

In this paper we introduce and develop the notion of minimal subspaces in the framework of algebraic and topological tensor product spaces. This mathematical structure arises in a natural way in the study of tensor representations. We use minimal subspaces to prove the existence of a best approximation, for any element in a Banach tensor space, by means of a tensor given in a typical representation format (Tucker, hierarchical, or tensor train). We show that this result holds in a tensor Banach space with a norm stronger than the injective norm and in an intersection of finitely many Banach tensor spaces satisfying some additional conditions. Examples using topological tensor products of standard Sobolev spaces are given.     

Tensor products of recurrent hypercyclic semigroups

We study tensor products of strongly continuous semigroups on Banach spaces that satisfy the hypercyclicity criterion, the recurrent hypercyclicity criterion or are chaotic.     

Power type asymptotically uniformly smooth and asymptotically uniformly flat norms

We provide a short characterization of p-asymptotic uniform smoothability and asymptotic uniform flatenability of operators and of Banach spaces. We use these characterizations to show that many asymptotic uniform smoothness properties pass to injective tensor products of operators and of Banach spaces. In particular, we prove that the injective tensor product of two asymptotically uniformly smooth Banach spaces is asymptotically uniformly smooth. We prove that for \(1<p<\infty \), the class of p-asymptotically uniformly smoothable operators can be endowed with an ideal norm making this class a Banach ideal. We also prove that the class of asymptotically uniformly flattenable operators can be endowed with an ideal norm making this class a Banach ideal.     

Atomic decompositions for tensor products and polynomial spaces

We study the existence of atomic decompositions for tensor products of Banach spaces and spaces of homogeneous polynomials. If a Banach space X admits an atomic decomposition of a certain kind, we show that the symmetrized tensor product of the elements of the atomic decomposition provides an atomic decomposition for the symmetric tensor product , for any symmetric tensor norm μ. In addition, the reciprocal statement is investigated and analogous consequences for the full tensor product are obtained. Finally we apply the previous results to establish the existence of monomial atomic decompositions for certain ideals of polynomials on X.     

Malliavin calculus and decoupling inequalities in Banach spaces

We develop a theory of Malliavin calculus for Banach space-valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Itô isometry to Banach spaces. In the white noise case we obtain two sided Lp-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Itô chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer's inequalities to give a new proof of a decoupling result for Gaussian chaoses in UMD Banach spaces.     

Polynomials on Banach lattices and positive tensor products

This paper is the first systematic study of homogeneous polynomials on Banach lattices. A variety of new Banach spaces and Banach lattices of multilinear maps, homogeneous polynomials, and operators are introduced. The main technique is to employ positive tensor products and quotients of positive tensor products. Our theorems generalize the results on orthogonally additive polynomials by Benyamini, Lassalle, and Llavona (2006) in [4], the results by Grecu and Ryan (2005) in [14], and the results by Sundaresan (1991) in [23].     

Existence of solutions for impulsive partial neutral functional differential equation with infinite delay

In this paper, the existence of mild solutions for first-order impulsive semilinear neutral functional differential equations with infinite delay in Banach spaces is investigated. We derive conditions in respect of the Hausdorff measure of noncompactness under which the mild solutions exist in Banach spaces. Our results improve and generalize some previous results.     

The Norm of a Complex Banach Lattice

In this note we study the problem how the complexification of a real Banach space can be normed in such a way that it becomes a complex Banach space from the point of view of the theory of cross-norms on tensor products of Banach spaces. In particular we show that the norm of a complex Banach lattice can be interpretated in terms of the l-tensor product of real Banach lattices.     

Weak compactness, transfinite Schauder bases and differentiability

We provide a characterization of the class of nonseparable Banach spaces that contain a nonseparable weakly compact set (respectively a relatively weakly compact transfinite basic sequence) in terms of differentiability properties of those spaces.     

Weyl's theorem, tensor products and multiplication operators

The transmission of “Weyl's theorem” from operators on Banach spaces to their tensor products, and also to their associated multiplication operators, is deconstructed.     

Banach frames for multivariate α-modulation spaces

The α-modulation spaces , α∈[0,1], form a family of spaces that include the Besov and modulation spaces as special cases. This paper is concerned with construction of Banach frames for α-modulation spaces in the multivariate setting. The frames constructed are unions of independent Riesz sequences based on tensor products of univariate brushlet functions, which simplifies the analysis of the full frame. We show that the multivariate α-modulation spaces can be completely characterized by the Banach frames constructed.     

Characterization of BMO in terms of rearrangement-invariant Banach function spaces

The atomic decomposition of Hardy spaces by atoms defined by rearrangement-invariant Banach function spaces is proved in this paper. Using this decomposition, we obtain the characterizations of BMO and Lipschitz spaces by rearrangement-invariant Banach function spaces. We also provide the sharp function characterization of the rearrangement-invariant Banach function spaces.     

Rational homotopy theory and differential graded category

We propose a generalization of Sullivan’s de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan’s theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions.     

Amenability of Banach algebras of compact operators

In this paper we study conditions on a Banach spaceX that ensure that the Banach algebraК(X) of compact operators is amenable. We give a symmetrized approximation property ofX which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. For direct sums this question is closely related to factorization of linear operators. In the final section we discuss some open questions, in particular, the converse problem of what properties ofX are implied by the amenability ofК(X). BEJ supported by MSRVP at Australian National University; GAW supported by SERC grant GR-F-74332.     

Minimal ideals of n-homogeneous polynomials on Banach spaces

The minimal kernel of a p-Banach ideal of n-homogeneous polynomials between Banach spaces is defined as a composition ideal, characterized to be the smallest ideal which coincides with the given one on finite-dimensional spaces and represented through tensor products with appropriate norms.     

Unconditionality in tensor products

It is proved that in order to study unconditional structures in tensor products of finite dimensional Banach spaces it is enough to consider a certain basis. This result is applied to spaces ofp-absolutely summing operators showing their “bad” structure.     

Algebraic reflexivity of the isometry group of some spaces of Lipschitz functions

We show that the isometry groups of Lip(X,d) and lip(X,dα) with α∈(0,1), for a compact metric space (X,d), are algebraically reflexive. We also prove that the sets of isometric reflections and generalized bi-circular projections on such spaces are algebraically reflexive. In order to achieve this, we characterize generalized bi-circular projections on these spaces.     

Antisymmetric tensor products of absolutely p-summing operators

We consider antisymmetric tensor products of absolutely p-summing operators. In connection with this second moments of determinants of random matrices appear. These second moments are closely related to approximation properties of the absolutely 2-summing operators and can be used to characterize some classes of infinite-dimensional Banach spaces. Finite-dimensional results are also obtained by this approach.     

Projections and averages of isometries on Lipschitz spaces

We characterize projections on spaces of Lipschitz functions expressed as the average of two and three linear surjective isometries. Generalized bi-circular projections are the only projections on these spaces given as the convex combination of two surjective isometries.