Raymond A. Ryan

Extendibility of homogeneous polynomials on Banach spaces

We study the n-homogeneous polynomials on a Banach space X that can be extended to any space containing X. We show that there is an upper bound on the norm of the extension. We construct a predual for the space of all extendible n-homogeneous polynomials on X and we characterize the extendible 2-homogeneous polynomials on X when X is a Hilbert space, an L1-space or an L∞-space.

Holomorphic mappings on

We describe the holomorphic mappings of bounded type, and the arbitrary holomorphic mappings from the complex Banach space 11 into a complex Banach space X. It is shown that these mappings have monomial expansions and the growth of the norms of the coefficients is characterized in each case. This characterization is used to give new descriptions of the compact open topology and the Nachbin ported topology on the space A((11; X) of holomorphic mappings, and to prove a lifting property for holomorphic mappings on 11. We also show that the monomials form an equicontinuous unconditional Schauder basis for the space ()((11), so) of holomorphic functions on 11 with the topology of uniform convergence on compact sets.

Zeros of polynomials on Banach spaces: The real story

Let E be a real Banach space. We show that either E admits a positive definite 2-homogeneous polynomial or every 2-homogeneous polynomial on E has an infinite dimensional subspace on which it is identically zero. Under addition assumptions, we show that such subspaces are non-separable. We examine analogous results for nuclear and absolutely (1,2)-summing 2-homogeneous polynomials and give necessary and sufficient conditions on a compact set K so that C(K) admits a positive definite 2-homogeneous polynomial or a positive definite nuclear 2-homogeneous polynomial.

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.

Uniform factorization for compact sets of operators

We prove a factorization result for relatively compact subsets of compact operators using the Bartle and Graves Selection Theorem, a characterization of relatively compact subsets of tensor products due to Grothendieck, and results of Figiel and Johnson on factorization of compact operators. A further proof, essentially based on the Banach-Dieudonné Theorem, is included. Our methods enable us to give an easier proof of a result of W.H. Graves and W.M. Ruess.

The Projective Tensor Product

In this chapter we investigate the simplest way to norm the tensor product of two Banach spaces. The projective tensor product linearizes bounded bilinear mappings just as the algebraic tensor product linearizes bilinear mappings. The projective tensor product derives its name from the fact that it behaves well with respect to quotient space constructions. The projective tensor product of l1 with X gives a representation of the space of absolutely summable sequences in X and projective tensor products with L(µ)lead to a study of the Bochner integral for Banach space valued functions. We also introduce the class of ℒ-spaces, whose finite dimensional structure is like that of l1. We study some techniques that make use of the Rademacher functions, including the Khinchine inequality. Finally, interpreting the elements of a projective tensor product as bilinear forms or operators leads to the introduction of the concept of nuclearity.

Tensor products of direct sums

A similar formula to the one established by Ansemil and Floret for symmetric tensor products of direct sums is proved for alternating and Jacobian tensor products. It is then applied to stable spaces where a number of isomorphisms between spaces of tensors or multilinear forms are unveiled. A connection between these problems and irreducible group representations is made.

THE NORM OF THE PRODUCT OF POLYNOMIALS IN INFINITE DIMENSIONS

Given a Banach space E and positive integers k and l we investigate the smallest constant C that satisfies ‖P‖‖Q‖ C‖PQ‖ for all k-homogeneous polynomials P and l-homogeneous polynomials Q on E. Our estimates are obtained using multilinear maps, the principle of local reflexivity and ideas from the geometry of Banach spaces (type and uniform convexity). We also examine the analogous problem for general polynomials on Banach spaces.

Positive polynomials on Riesz spaces

We prove some properties of positive polynomial mappings between Riesz spaces, using finite difference calculus. We establish the polynomial analogue of the classical result that positive, additive mappings are linear. And we prove a polynomial version of the Kantorovich extension theorem.

The Approximation Property

In this chapter we introduce the approximation property for Banach spaces. The possession of this property leads to the resolution of several outstanding issues concerning projective and injective tensor products. We then consider the following question: when are the projective or injective tensor products of reflexive spaces themselves reflexive? A satisfactory answer requires the use of the approximation property. Finally, we study tensor products of Banach spaces with Schauder bases.