Homogeneous Orthogonally Additive Polynomials on Banach Lattices

The main result in this paper is a representation theorem forhomogeneous orthogonally additive polynomials on Banach lattices.The representation theorem is used to study the linear spanof the set of zeros of homogeneous real-valued orthogonallyadditive polynomials. It is shown that in certain lattices everyelement can be represented as the sum of two or three zerosor, at least, can be approximated by such sums. It is also indicatedhow these results can be used to study weak topologies inducedby orthogonally additive polynomials on Banach lattices. 2000Mathematics Subject Classification 46G25, 46B42, 47B38.     

Polynomials on Banach spaces with unconditional bases

We study the classes of homogeneous polynomials on a Banach space with unconditional Schauder basis that have unconditionally convergent monomial expansions relative to this basis. We extend some results of Matos, and we show that the homogeneous polynomials with unconditionally convergent expansions coincide with the polynomials that are regular with respect to the Banach lattices structure of the domain.

     

On compact domination of homogeneous orthogonally additive polynomials

We prove an analog of the Dodds–Fremlin–Wickstead Theorem on compact domination for homogeneous orthogonally additive polynomials in Banach lattices. The proof is based on linearization of the polynomials which was established earlier by the author.     

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.     

A polynomial Hutton theorem with applications

Extending a classical linear result due to Hutton to a nonlinear setting, we prove that a continuous homogeneous polynomial between Banach spaces can be approximated by finite rank polynomials if and only if its adjoint can be approximated by finite rank linear operators. Among other consequences, we apply this result to generalize a classical result due to Aron and Schottenloher about the approximation property on spaces of polynomials and a recent result due to Çaliskan and Rueda about the quasi-approximation property on projective symmetric tensor products.     

Regularity of Continuous Multilinear Operators and Homogeneous Polynomials

Mathematical Notes - Regular multilinear operators and regular homogeneous polynomials acting between Banach lattices are automatically continuous, but the converse, in general, is not true. The...     

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 spectral synthesis in commutative Banach algebras. II

Let A be a semisimple and regular commutative Banach algebra with structure space Δ(A). Continuing our investigation in [E. Kaniuth, Weak spectral synthesis in commutative Banach algebras, J. Funct. Anal. 254 (2008) 987-1002], we establish various results on intersections and unions of weak spectral sets and weak Ditkin sets in Δ(A). As an important example, the algebra of n-times continuously differentiable functions is studied in detail. In addition, we prove a theorem on spectral synthesis for projective tensor products of commutative Banach algebras which applies to Fourier algebras of locally compact groups.     

Invariant sublattices for positive operators

There are, by now, many results which guarantee that positive operators on Banach lattices have non-trivial closed invariant sublattices. In particular, this is true for every positive compact operator. Apart from some results of a general nature, in this paper we present several examples of positive operators on Banach lattices which do not have non-trivial closed invariant sublattices. These examples include both AM-spaces and Banach lattices with an order continuous norm and which are and are not atomic. In all these cases we can ensure that the operators do possess non-trivial closed invariant subspaces.     

On the polynomial Lindenstrauss theorem

Under certain hypotheses on the Banach space X, we show that the set of N-homogeneous polynomials from X to any dual space, whose Aron–Berner extensions are norm attaining, is dense in the space of all continuous N-homogeneous polynomials. To this end we prove an integral formula for the duality between tensor products and polynomials. We also exhibit examples of Lorentz sequence spaces for which there is no polynomial Bishop–Phelps theorem, but our results apply. Finally we address quantitative versions, in the sense of Bollobás, of these results.     

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.     

Functional completions of Archimedean vector lattices

We study completions of Archimedean vector lattices relative to any nonempty set of positively homogeneous functions on finite-dimensional real vector spaces. Examples of such completions include square mean closed and geometric mean closed vector lattices, amongst others. These functional completions also lead to a universal definition of the complexification of any Archimedean vector lattice and a theory of tensor products and powers of complex vector lattices in a companion paper.     

Orthogonalkompakte Teilmengen topologischer Vektorverbände

The criterion of Dunford-Pettis for weak compactness in Banach lattices of L¹() type can be derived from a characterisation of weak sequentially complete topological vector lattices. This can be done by introducing a concept which reduces to uniform integrability in the L¹() case ([1], [8]). In other cases suitable choice of the topology leads to definitions given by [4], [9], [11] and [12]. It is shown in this paper that the orthogonally compact subsets of a Banach lattice are characterized as those relatively weakly compact sets on which the norm and the order topology agree.

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Der Inhalt dieser Arbeit ist ein Auszug aus der Dissertation des Autors an der Universität Dortmund     

Operator Newton polynomials and well-solvable problems for generalized Euler equation

The study of well-solvable operator equations in a Banach space, which was initiated by the authors in [4, 5], is continued. Namely, it is proved by means of Maslov’s operator method that a polynomial equation with abstract Newton polynomials is well solvable in the sense of Hadamard. The obtained results are applied to prove that a large class of problems for differential equations with variable coefficient having a singularity (such equations are called generalized Euler equations in the paper) are well solvable.     

Normed tensor products of Banach lattices

On the tensor productE⊗F of a pair of order complete Banach lattices, two cross norms (called thel-andm-norm, respectively) are introduced. These cross-norms (which depend on the order of factors, and are permuted when the latter is inverted) have the property that the respective completions ofE⊗F are Banach lattices under the ordering defined by the closure of the projective cone. Moreover, they are self-dual with respect to <E ⊗F, E’> and coincide with well-known tensor norms in important special cases.     

Jensen type inequalities for positive bilinear operators

A general Jensen type inequality for positive bilinear operators between uniformly complete vector lattice is proved. Then some new inequalities for linear and bilinear operators and an interpolation result for positive bilinear operators between Calderón–Lozanovskiĭ spaces are deduced. The proof of the main result relies upon homogeneous functional calculus on vector lattices and the Fremlin tensor product of Archimedean vector lattices.     

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     

An Embedding Theorem for Pseudoconvex Domains in Banach Spaces

We show that a pseudoconvex open subset of a Banach space with an unconditional basis is biholomorphic to a closed direct submanifold of a Banach space with an unconditional basis.This research supported in part by NSF grant DMS-0203072 and a Bilsland Dissertation Fellowship from the Graduate School of Purdue University. This paper is the result of my thesis research under the direction of László Lempert, and I would like to thank him for his patient and wise guidance. I would also like to thank Andreas Defant and David Pérez-García for their suggestions regarding tensor products and directing me to the paper [4], and Steve Bell for his suggestions.     

Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces

Recently, in the article [LW], the authors use the notion of polynomials in metric spaces of homogeneous type (in the sense of Coifman-Weiss) to prove a relationship between high order Poincaré inequalities and representation formulas involving fractional integrals of high order, assuming only that is a doubling measure and that geodesics exist. Motivated by this and by recent work in [H], [FHK], [KS] and [FLW] about first order Sobolev spaces in metric spaces, we define Sobolev spaces of high order in such metric spaces . We prove that several definitions are equivalent if functions of polynomial type exist. In the case of stratified groups, where polynomials do exist, we show that our spaces are equivalent to the Sobolev spaces defined by Folland and Stein in [FS]. Our results also give some alternate definitions of Sobolev spaces in the classical Euclidean case. Received: 10 February 1999 / Published online: 1 February 2002     

Centralisers of spaces of symmetric tensor products and applications

We show that the centraliser of the space of n-fold symmetric injective tensors, n≥2, on a real Banach space is trivial. With a geometric condition on the set of extreme points of its dual, the space of integral polynomials we obtain the same result for complex Banach spaces. We give some applications of this results to centralisers of spaces of homogeneous polynomials and complex Banach spaces. In addition, we derive a Banach-Stone Theorem for spaces of vector-valued approximable polynomials. This project was supported in part by Enterprise Ireland, International Collaboration Grant – 2004 (IC/2004/009). The second author was also partially supported by PIP 5272,UBACYTX108 and PICT 03-15033