Unconditional Basis and Gordon–Lewis Constants for Spaces of Polynomials

Abstract

No infinite dimensional Banach space X is known which has the property that for m 2 the Banach space of all continuous m-homogeneous polynomials on X has an unconditional basis. Following a program originally initiated by Gordon and Lewis we study unconditionality in spaces of m-homogeneous polynomials and symmetric tensor products of order m in Banach spaces. We show that for each Banach space X which has a dual with an unconditional basis (x i *), the approximable (nuclear) m-homogeneous polynomials on X have an unconditional basis if and only if the monomial basis with respect to (x i *) is unconditional. Moreover, we determine an asymptotically correct estimate for the unconditional basis constant of all m-homogeneous polynomials on l n p and use this result to narrow down considerably the list of natural candidates X with the above property. Academic Press

Schauder Bases for Symmetric Tensor Products

“…Working for tensors product of two spaces E ⊗ F with a uniform crossnorm, Pisier [9] and Schütt [12] obtained independently that the tensor product basis is unconditional if and only if E⊗ µ F has the Gordon-Lewis property. Their results have been extended to n-fold symmetric and full tensor products by Defant, Díaz, Garcia and Maestre [4]. Considering the dual problem, that is the spaces P w ( n E) and P( n E) having unconditional basis, they narrowed down considerably the list of candidates E with this property, showing that it does not happen for any Banach space that contains uniformly complemented one of the sequences ( n p ) n∈N with 1 < p ≤ ∞.…”

“…The dual problem, whether the monomials are a basis in the space of homogeneous polynomials, was dealt with by Dimant in her thesis [1], as well as in two other articles, together with Dineen [2] and Zalduendo [3]. The unconditionality (or lack thereof) of the monomial basis was extensively analysed by Defant, Díaz, Garcia and Maestre [4]. …”

Unconditional bases in tensor products of Hilbert spaces

“…In the context of the theory of polynomials in Banach spaces, the problem of finding unconditional bases in symmetric tensor products has attracted recently a lot of attention (see for instance [7] or [10] and the references therein). The symmetric case of Kwapień and Pełczyński's result appeared in [6], only for the case of two spaces.…”

A tensor norm preserving unconditionality for $\mathcal{L}_p$-spaces

“…Our motivation mainly comes from the multilinear theory of Banach spaces. In particular, from a recent positive answer to a conjecture of Dineen [16], the space of m-homogeneous polynomials on an infinite dimensional Banach space never has an unconditional basis (see [11] and also [8]). In [9] the authors apply unconditionality in tensor products to give multidimensional analogues of Bohr's classical power series theorem from [5] (for more on that see [1], [3], [4], [9], [17]).…”