Anti-Wick Symbols for Infinite Products in K-Homology

We consider infinite products in K-homology. We study these products in relation with operators on filtered Hilbert spaces, and infinite iterations of universal constructions on C*-algebras. In particular, infinite tensor power of extensions of pseudodifferential operators on R are considered. We extend anti-Wick pseudodifferential operators to infinite tensor products of spaces of the type L ²(R), and compare our infinite tensor power construction with an extension of pseudodifferential operators on R . We show that the K-theory connecting maps coincide. We propose a natural definition of ellipticity for anti-Wick operators on R, compute the corresponding index, and draw some consequences concerning these operators.     

Tensor Product Weight Representations of the Neveu–Schwarz Algebra

We study the tensor product of a highest weight module with an intermediate series module over the Neveu–Schwarz algebra. If the highest weight module is nontrivial, the weight spaces of such a tensor product are infinite dimensional. We show that such a tensor product is indecomposable. Using a “shifting technique” developed by H. Chen, X. Guo, and K. Zhao for the Virasoro algebra case, we give necessary and sufficient conditions for such a tensor product to be irreducible. Furthermore, we give necessary and sufficient conditions for two such tensor products to be isomorphic.     

Multirectangular invariants for power Köthe spaces

Using some new linear topological invariants, isomorphisms and quasidiagonal isomorphisms are investigated on the class of first type power Köthe spaces [Proceedings of 7th Winter School in Drogobych, 1976, pp. 101-126; Turkish J. Math. 20 (1996) 237-289; Linear Topol. Spaces Complex Anal. 2 (1995) 35-44]. This is the smallest class of Köthe spaces containing all Cartesian and projective tensor products of power series spaces and closed with respect to taking of basic subspaces (closed linear hulls of subsets of the canonical basis). As an application, it is shown that isomorphic spaces from this class have, up to quasidiagonal isomorphisms, the same basic subspaces of finite (infinite) type.     

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.     

Quasiregular representations of the infinite-dimensional nilpotent group

In the present work an analog of the quasiregular representation which is well known for locally-compact groups is constructed for the nilpotent infinite-dimensional group and a criterion for its irreducibility is presented. This construction uses the infinite tensor product of arbitrary Gaussian measures in the spaces Rm with m>1 extending in a rather subtle way previous work of the second author for the infinite tensor product of one-dimensional Gaussian measures.     

A General Multiresolution Method for Fitting Functions on the Sphere

In [7], Lyche and Schumaker have described a method for fitting functions of class C ¹ on the sphere which is based on tensor products of quadratic polynomial splines and trigonometric splines of order three associated with uniform knots. In this paper, we present a multiresolution method leading to C ²-functions on the sphere, using tensor products of polynomial and trigonometric splines of odd order with arbitrary simple knot sequences. We determine the decomposition and reconstruction matrices corresponding to the polynomial and trigonometric spline spaces. We describe the general tensor product decomposition and reconstruction algorithms in matrix form which are convenient for the compression of surfaces. We give the different steps of the computer implementation of these algorithms and, finally, we present a test example.     

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.     

Projections on tensor products of Banach spaces

We characterize norm hermitian operators on classes of tensor products of Banach spaces and derive results for the particular settings of injective and projective tensor products. We provide a characterization of bi-circular and generalized bi-circular projections on tensor products of Banach spaces supporting only dyadic surjective isometries. Received: 26 February 2007, Revised: 30 May 2007     

Best m-Term Approximation and Sobolev–Besov Spaces of Dominating Mixed Smoothness—the Case of Compact Embeddings

We shall investigate the asymptotic behavior of the widths of best m-term approximation with respect to tensor products of Sobolev as well as Besov spaces in case of compact embeddings. Furthermore, we compare best m-term approximation with optimal linear approximation and entropy numbers.     

Essential normality of some classes of operators in infinite tensor products of Hilbert spaces

The essential normality of some classes of operators in stabilized infinite tensor products of Hilbert spaces is established by using the general criteria for the domains of definition of spectral integrals to be essential.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1164–1170, September, 1994.     

Tensor product representation of Köthe-Bochner spaces and their dual spaces

We provide a tensor product representation of Köthe-Bochner function spaces of vector valued integrable functions. As an application, we show that the dual space of a Köthe-Bochner function space can be understood as a space of operators satisfying a certain extension property. We apply our results in order to give an alternate representation of the dual of the Bochner spaces of p-integrable functions and to analyze some properties of the natural norms \(\Delta _p\) that are defined on the associated tensor products.     

Productivity numbers in topological spaces

The κ-productivity of classes C of topological spaces closed under quotients and disjoint sums is characterized by means of Cantor spaces. The smallest infinite cardinals κ such that such classes are not κ-productive are submeasurable cardinals. It follows that if a class of topological spaces is closed under quotients, disjoint sums and countable products, it is closed under products of non-sequentially many spaces (thus under all products, if sequential cardinals do not exist).     

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.     

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.     

Tractability through increasing smoothness

We prove that some multivariate linear tensor product problems are tractable in the worst case setting if they are defined as tensor products of univariate problems with logarithmically increasing smoothness. This is demonstrated for the approximation problem defined over Korobov spaces and for the approximation problem of certain diagonal operators. For these two problems we show necessary and sufficient conditions on the smoothness parameters of the univariate problems to obtain strong polynomial tractability. We prove that polynomial tractability is equivalent to strong polynomial tractability, and that weak tractability always holds for these problems. Under a mild assumption, the Korobov space consists of periodic functions. Periodicity is crucial since the approximation problem defined over Sobolev spaces of non-periodic functions with a special choice of the norm is not polynomially tractable for all smoothness parameters no matter how fast they go to infinity. Furthermore, depending on the choice of the norm we can even lose weak tractability.     

When does the Haver property imply selective screenability?

We point out that in metric spaces Haver's property is not equivalent to the property introduced by Addis and Gresham. We prove that they are equal when the space has the Hurewicz property. We prove several results about the preservation of Haver's property in products. We show that if a separable metric space has the Haver property, and the nth power has the Hurewicz property, then the nth power has the Addis-Gresham property. R. Pol showed earlier that this is not the case when the Hurewicz property is replaced by the weaker Menger property. We introduce new classes of weakly infinite dimensional spaces.     

Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product

We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the null spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares (?) solutions to a multilinear system and establish the relationship between the minimum-norm (N) least-squares (?) solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.     

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.