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 ℓnp and use this result to narrow down considerably the list of natural candidates X with the above property.