Polynomial properties and symmetric tensor product of Banach spaces

Abstract

We study some classes of distinguished subsets of a Banach space in terms of polynomials and their relationship. This allows us to develop a systematic approach to study polynomial properties on a Banach space. We apply this approach to obtain several known and new results on the symmetric tensor product of a Banach space in a unified way.

Polynomial sequential continuity on C(K,E) spaces

“…By the polarization identity, they are equivalent to the fact that every weakly Cauchy (or weakly convergent) sequence in E is τ M -convergent. In [7], this property is called the M-Sequential Continuity property. In [10], a space with property SP 1,M is called an M Mspace, and a space with property P 1,M is called a P M -space.…”

“…e) For N ∈ N, a space is said to have the P ≤N Dunford-Pettis property (P ≤N DP) if, for each τ ≤N null sequence (x n ) ⊂ X and every weakly null sequence (P n ) ⊂ P( N X), we have that P n (x n ) converges to 0. This property was defined in [15] and further studied in [6] (see also [7] for a related notion). A Banach space is said to have the P-Dunford-Pettis property (P-DP) if, for each m ∈ N, for every weakly null sequence (P n ) ⊂ P( m X) and for every τ ≤∞ null sequence (x n ) ⊂ X, we have that P n (x n ) converges to 0.…”

An invariant subspace problem for multilinear operators on finite dimensional spaces

“…, v) for all v ∈ E). Let P( m E; E) denote the Banach space of mhomogeneous polynomials, see [8]. A book by Dineen [12] gives an informative and exhaustive literature on polynomial operators and their relationship with multilinear operators.…”

“…× E → F is called an m-linear map if it is linear in each of the m variables. We denote the space of such continuous maps by L( m E; F ), see [8].…”